Section 6.8 - Variation

Four types of variation in this section:

Four types of variation in this section:

1. Direct Variation

Four types of variation in this section:

1. Direct Variation

2. Inverse Variation

Four types of variation in this section:

1. Direct Variation

2. Inverse Variation

3. Combined Variation (really just a combination of #1 and

#2)

Four types of variation in this section:

1. Direct Variation

2. Inverse Variation

3. Combined Variation (really just a combination of #1 and

#2)

4. Joint Variation

Direct Variation  The relationship “y varies directly as x” or “y is proportional to x” could be written in equation form as:

Direct Variation  The relationship “y varies directly as x” or “y is proportional to x” could be written in equation form as:

y= k x

Direct Variation  The relationship “y varies directly as x” or “y is proportional to x” could be written in equation form as:

where k represents the constant of proportionality

y= k x

Direct Variation  The relationship “y varies directly as x” or “y is proportional to x” could be written in equation form as:

where k represents the constant of proportionality

y= k xExample The circumference,C, of a circle varies directly to the diameter, d, of the circle as seen in the equation:

Direct Variation  The relationship “y varies directly as x” or “y is proportional to x” could be written in equation form as:

where k represents the constant of proportionality

y= k xExample

The circumference,C, of a circle varies directly to the diameter, d, of the circle as seen in the equation: C= π d

Direct Variation  The relationship “y varies directly as x” or “y is proportional to x” could be written in equation form as:

where k represents the constant of proportionality

y= k xExample

The circumference,C, of a circle varies directly to the diameter, d, of the circle as seen in the equation:

In this case the constant of proportionality is the number π ≈ 3.14

C= π d

So why call it direct variation?

So why call it direct variation?

Look at the circle below. What will happen to the circle’s circumference (distance around) if we make the diameter(distance across) bigger?

d = 5 inchesC = π(5) ≈ 15.7 in.

So why call it direct variation?

Look at the circle below. What will happen to the circle’s circumference (distance around) if we make the diameter(distance across) bigger?

d = 5 inchesC = π(5) ≈ 15.7 in.

So why call it direct variation?

Look at the circle below. What will happen to the circle’s circumference (distance around) if we make the diameter(distance across) bigger?

d = 5 inchesC = π(5) ≈ 15.7 in.

d = 8 inchesC = π(8) ≈ 25.12 in.

So why call it direct variation?

Look at the circle below. What will happen to the circle’s circumference (distance around) if we make the diameter(distance across) bigger?

Notice that increasing the diameter also increased the circumference. This is typical of things that vary directly. Increasing one variable will also increase the other variable. Likewise decreasing one variable will also decrease the other variable. This is why it is called direct variation, whatever you do to one variable (increase or decrease) it will directly affect the other variable in the same way.

d = 5 inchesC = π(5) ≈ 15.7 in.

d = 8 inchesC = π(8) ≈ 25.12 in.

Exercise 1: Write a general equation that represents the given relationship.

a. The electric resistance, R, of a wire varies directly as its length, L.

b. The volume, V, of a sphere varies directly as the cube of its radius.

Exercise 1: Write a general equation that represents the given relationship.

a. The electric resistance, R, of a wire varies directly as its length, L.

b. The volume, V, of a sphere varies directly as the cube of its radius.

Exercise 1: Write a general equation that represents the given relationship.

a. The electric resistance, R, of a wire varies directly as its length, L.

b. The volume, V, of a sphere varies directly as the cube of its radius.

Remember: direct variation always looks like y = kx. However, we now have different variables.

Exercise 1: Write a general equation that represents the given relationship.

a. The electric resistance, R, of a wire varies directly as its length, L.

b. The volume, V, of a sphere varies directly as the cube of its radius.

Remember: direct variation always looks like y = kx. However, we now have different variables.R = k L

Exercise 1: Write a general equation that represents the given relationship.

a. The electric resistance, R, of a wire varies directly as its length, L.

b. The volume, V, of a sphere varies directly as the cube of its radius.

Remember: direct variation always looks like y = kx. However, we now have different variables.R = k L

Exercise 1: Write a general equation that represents the given relationship.

a. The electric resistance, R, of a wire varies directly as its length, L.

b. The volume, V, of a sphere varies directly as the cube of its radius.

Remember: direct variation always looks like y = kx. However, we now have different variables.R = k L

Exercise 1: Write a general equation that represents the given relationship.

a. The electric resistance, R, of a wire varies directly as its length, L.

b. The volume, V, of a sphere varies directly as the cube of its radius.

Remember: direct variation always looks like y = kx. However, we now have different variables.R = k LV = k r 3

Exercise 2:

a. Write a general equation that represents the following relationship.When you swim underwater the pressure, P, in your ears varies directly as the depth, d, at which you are swimming.

Exercise 2:

a. Write a general equation that represents the following relationship.When you swim underwater the pressure, P, in your ears varies directly as the depth, d, at which you are swimming.

P = kd

Exercise 2:

a. Write a general equation that represents the following relationship.When you swim underwater the pressure, P, in your ears varies directly as the depth, d, at which you are swimming.

P = kd

b. Suppose you know that at a depth of 20 ft the pressure in your ears is 8.6 pounds per square inch. Calculate the constant of proportionality, k.

Exercise 2:

a. Write a general equation that represents the following relationship.When you swim underwater the pressure, P, in your ears varies directly as the depth, d, at which you are swimming.

P = kd

b. Suppose you know that at a depth of 20 ft the pressure in your ears is 8.6 pounds per square inch. Calculate the constant of proportionality, k.

P = 8.6

Exercise 2:

a. Write a general equation that represents the following relationship.When you swim underwater the pressure, P, in your ears varies directly as the depth, d, at which you are swimming.

P = kd

b. Suppose you know that at a depth of 20 ft the pressure in your ears is 8.6 pounds per square inch. Calculate the constant of proportionality, k.

P = 8.6 and d = 20

Exercise 2:

a. Write a general equation that represents the following relationship.When you swim underwater the pressure, P, in your ears varies directly as the depth, d, at which you are swimming.

P = kd

b. Suppose you know that at a depth of 20 ft the pressure in your ears is 8.6 pounds per square inch. Calculate the constant of proportionality, k.

P = 8.6 and d = 20 So…

Exercise 2:

a. Write a general equation that represents the following relationship.When you swim underwater the pressure, P, in your ears varies directly as the depth, d, at which you are swimming.

P = kd

b. Suppose you know that at a depth of 20 ft the pressure in your ears is 8.6 pounds per square inch. Calculate the constant of proportionality, k.

P = 8.6 and d = 20 So…

8.6 (20)k

Exercise 2:

a. Write a general equation that represents the following relationship.When you swim underwater the pressure, P, in your ears varies directly as the depth, d, at which you are swimming.

P = kd

b. Suppose you know that at a depth of 20 ft the pressure in your ears is 8.6 pounds per square inch. Calculate the constant of proportionality, k.

P = 8.6 and d = 20 So…

8.6 (20)8.6 20

kk

Exercise 2:

a. Write a general equation that represents the following relationship.When you swim underwater the pressure, P, in your ears varies directly as the depth, d, at which you are swimming.

P = kd

b. Suppose you know that at a depth of 20 ft the pressure in your ears is 8.6 pounds per square inch. Calculate the constant of proportionality, k.

P = 8.6 and d = 20 So…

8.6 (20)8.6 20

8.6 2020

kk

20k

Exercise 2:

a. Write a general equation that represents the following relationship.When you swim underwater the pressure, P, in your ears varies directly as the depth, d, at which you are swimming.

P = kd

b. Suppose you know that at a depth of 20 ft the pressure in your ears is 8.6 pounds per square inch. Calculate the constant of proportionality, k.

P = 8.6 and d = 20 So…

8.6 (20)8.6 20

8.6 2020

kk

20k

0.43k

Exercise 2:

a. Write a general equation that represents the following relationship.When you swim underwater the pressure, P, in your ears varies directly as the depth, d, at which you are swimming.

P = kd

b. Suppose you know that at a depth of 20 ft the pressure in your ears is 8.6 pounds per square inch. Calculate the constant of proportionality, k.

P = 8.6 and d = 20 So…

8.6 (20)8.6 20

8.6 2020

kk

20k

0.43k

c. At an underwater depth of 80 ft, what is the pressure in your ears?

Exercise 2:

a. Write a general equation that represents the following relationship.When you swim underwater the pressure, P, in your ears varies directly as the depth, d, at which you are swimming.

P = kd

b. Suppose you know that at a depth of 20 ft the pressure in your ears is 8.6 pounds per square inch. Calculate the constant of proportionality, k.

P = 8.6 and d = 20 So…

8.6 (20)8.6 20

8.6 2020

kk

20k

0.43k

c. At an underwater depth of 80 ft, what is the pressure in your ears?We now know(from part b) that when you are underwater the constant of proportionality is k = 0.43. So our actual equation is

Exercise 2:

a. Write a general equation that represents the following relationship.When you swim underwater the pressure, P, in your ears varies directly as the depth, d, at which you are swimming.

P = kd

b. Suppose you know that at a depth of 20 ft the pressure in your ears is 8.6 pounds per square inch. Calculate the constant of proportionality, k.

P = 8.6 and d = 20 So…

8.6 (20)8.6 20

8.6 2020

kk

20k

0.43k

c. At an underwater depth of 80 ft, what is the pressure in your ears?We now know(from part b) that when you are underwater the constant of proportionality is k = 0.43. So our actual equation is P = 0.43d

Exercise 2:

a. Write a general equation that represents the following relationship.When you swim underwater the pressure, P, in your ears varies directly as the depth, d, at which you are swimming.

P = kd

b. Suppose you know that at a depth of 20 ft the pressure in your ears is 8.6 pounds per square inch. Calculate the constant of proportionality, k.

P = 8.6 and d = 20 So…

8.6 (20)8.6 20

8.6 2020

kk

20k

0.43k

c. At an underwater depth of 80 ft, what is the pressure in your ears?We now know(from part b) that when you are underwater the constant of proportionality is k = 0.43. So our actual equation is P = 0.43d

At a depth of 80 ft we would have 0.43(80)P

Exercise 2:

a. Write a general equation that represents the following relationship.When you swim underwater the pressure, P, in your ears varies directly as the depth, d, at which you are swimming.

P = kd

b. Suppose you know that at a depth of 20 ft the pressure in your ears is 8.6 pounds per square inch. Calculate the constant of proportionality, k.

P = 8.6 and d = 20 So…

8.6 (20)8.6 20

8.6 2020

kk

20k

0.43k

c. At an underwater depth of 80 ft, what is the pressure in your ears?We now know(from part b) that when you are underwater the constant of proportionality is k = 0.43. So our actual equation is P = 0.43d

At a depth of 80 ft we would have 0.43(80) 34.4P

Exercise 2:

a. Write a general equation that represents the following relationship.When you swim underwater the pressure, P, in your ears varies directly as the depth, d, at which you are swimming.

P = kd

b. Suppose you know that at a depth of 20 ft the pressure in your ears is 8.6 pounds per square inch. Calculate the constant of proportionality, k.

P = 8.6 and d = 20 So…

8.6 (20)8.6 20

8.6 2020

kk

20k

0.43k

c. At an underwater depth of 80 ft, what is the pressure in your ears?We now know(from part b) that when you are underwater the constant of proportionality is k = 0.43. So our actual equation is P = 0.43d

At a depth of 80 ft we would have 0.43(80) 34.4P

At a depth of 80 ft the pressure in your ears would be 34.4 pounds per sq. inch.

The three parts of exercise 2 represent a typical problem in this section.

Most problems follow the same basic pattern (see Blitzer textbook, page 465):

The three parts of exercise 2 represent a typical problem in this section.

Most problems follow the same basic pattern (see Blitzer textbook, page 465):

1. Write an equation that models the statement.

The three parts of exercise 2 represent a typical problem in this section.

Most problems follow the same basic pattern (see Blitzer textbook, page 465):

1. Write an equation that models the statement.

2. Substitute the given values in to the equation to find the value of k.

The three parts of exercise 2 represent a typical problem in this section.

Most problems follow the same basic pattern (see Blitzer textbook, page 465):

1. Write an equation that models the statement.

2. Substitute the given values in to the equation to find the value of k.

3. Substitute the value of k into the equation from step 1.

The three parts of exercise 2 represent a typical problem in this section.

Most problems follow the same basic pattern (see Blitzer textbook, page 465):

1. Write an equation that models the statement.

2. Substitute the given values in to the equation to find the value of k.

3. Substitute the value of k into the equation from step 1.

4. Use the equation from step 3 to answer the problem’s question.

Exercise 3:

The height that a ball bounces varies directly as the height from which it was dropped. A tennis ball dropped from 12 inches bounces 8.4 inches. From what height was the tennis ball dropped if it bounces 56 inches?

Exercise 3:

The height that a ball bounces varies directly as the height from which it was dropped. A tennis ball dropped from 12 inches bounces 8.4 inches. From what height was the tennis ball dropped if it bounces 56 inches?Step 1: Write the equation.

Exercise 3:

The height that a ball bounces varies directly as the height from which it was dropped. A tennis ball dropped from 12 inches bounces 8.4 inches. From what height was the tennis ball dropped if it bounces 56 inches?Step 1: Write the equation.

Exercise 3:

The height that a ball bounces varies directly as the height from which it was dropped. A tennis ball dropped from 12 inches bounces 8.4 inches. From what height was the tennis ball dropped if it bounces 56 inches?Step 1: Write the equation.

Exercise 3:

The height that a ball bounces varies directly as the height from which it was dropped. A tennis ball dropped from 12 inches bounces 8.4 inches. From what height was the tennis ball dropped if it bounces 56 inches?Step 1: Write the equation. Let B be the height the ball bounces and d the height from which the ball was dropped.

Exercise 3:

The height that a ball bounces varies directly as the height from which it was dropped. A tennis ball dropped from 12 inches bounces 8.4 inches. From what height was the tennis ball dropped if it bounces 56 inches?Step 1: Write the equation. Let B be the height the ball bounces and d the height from which the ball was dropped.

B kd

Exercise 3:

The height that a ball bounces varies directly as the height from which it was dropped. A tennis ball dropped from 12 inches bounces 8.4 inches. From what height was the tennis ball dropped if it bounces 56 inches?Step 1: Write the equation. Let B be the height the ball bounces and d the height from which the ball was dropped.

B kdStep 2: Substitute the givens to find k.

Exercise 3:

The height that a ball bounces varies directly as the height from which it was dropped. A tennis ball dropped from 12 inches bounces 8.4 inches. From what height was the tennis ball dropped if it bounces 56 inches?Step 1: Write the equation. Let B be the height the ball bounces and d the height from which the ball was dropped.

B kdStep 2: Substitute the givens to find k. When 12 in. the ball bounces 8.4 in.d B

Exercise 3:

The height that a ball bounces varies directly as the height from which it was dropped. A tennis ball dropped from 12 inches bounces 8.4 inches. From what height was the tennis ball dropped if it bounces 56 inches?Step 1: Write the equation. Let B be the height the ball bounces and d the height from which the ball was dropped.

B kdStep 2: Substitute the givens to find k. When 12 in. the ball bounces 8.4 in.d B

8.4 (12) 8.4 12k k

Exercise 3:

The height that a ball bounces varies directly as the height from which it was dropped. A tennis ball dropped from 12 inches bounces 8.4 inches. From what height was the tennis ball dropped if it bounces 56 inches?Step 1: Write the equation. Let B be the height the ball bounces and d the height from which the ball was dropped.

B kdStep 2: Substitute the givens to find k. When 12 in. the ball bounces 8.4 in.d B

8.4 (12) 8.4 128.4 1212 12

k kk

Exercise 3:

The height that a ball bounces varies directly as the height from which it was dropped. A tennis ball dropped from 12 inches bounces 8.4 inches. From what height was the tennis ball dropped if it bounces 56 inches?Step 1: Write the equation. Let B be the height the ball bounces and d the height from which the ball was dropped.

B kdStep 2: Substitute the givens to find k. When 12 in. the ball bounces 8.4 in.d B

8.4 (12) 8.4 12

8.4 1212

k k

12k

Exercise 3:

The height that a ball bounces varies directly as the height from which it was dropped. A tennis ball dropped from 12 inches bounces 8.4 inches. From what height was the tennis ball dropped if it bounces 56 inches?Step 1: Write the equation. Let B be the height the ball bounces and d the height from which the ball was dropped.

B kdStep 2: Substitute the givens to find k. When 12 in. the ball bounces 8.4 in.d B

8.4 (12) 8.4 12

8.4 1212

k k

12k

0.7k

Exercise 3:

The height that a ball bounces varies directly as the height from which it was dropped. A tennis ball dropped from 12 inches bounces 8.4 inches. From what height was the tennis ball dropped if it bounces 56 inches?Step 1: Write the equation. Let B be the height the ball bounces and d the height from which the ball was dropped.

B kdStep 2: Substitute the givens to find k. When 12 in. the ball bounces 8.4 in.d B

8.4 (12) 8.4 12

8.4 1212

k k

12k

0.7k Step 3: Substitute k into the equation from step 1.

Exercise 3:

The height that a ball bounces varies directly as the height from which it was dropped. A tennis ball dropped from 12 inches bounces 8.4 inches. From what height was the tennis ball dropped if it bounces 56 inches?Step 1: Write the equation. Let B be the height the ball bounces and d the height from which the ball was dropped.

B kdStep 2: Substitute the givens to find k. When 12 in. the ball bounces 8.4 in.d B

8.4 (12) 8.4 12

8.4 1212

k k

12k

0.7k Step 3: Substitute k into the equation from step 1. 0.7B d

Exercise 3:

The height that a ball bounces varies directly as the height from which it was dropped. A tennis ball dropped from 12 inches bounces 8.4 inches. From what height was the tennis ball dropped if it bounces 56 inches?Step 1: Write the equation. Let B be the height the ball bounces and d the height from which the ball was dropped.

B kdStep 2: Substitute the givens to find k. When 12 in. the ball bounces 8.4 in.d B

8.4 (12) 8.4 12

8.4 1212

k k

12k

0.7k Step 3: Substitute k into the equation from step 1. 0.7B dStep 4: Use the equation from step 3 to answer the question.

Exercise 3:

The height that a ball bounces varies directly as the height from which it was dropped. A tennis ball dropped from 12 inches bounces 8.4 inches. From what height was the tennis ball dropped if it bounces 56 inches?Step 1: Write the equation. Let B be the height the ball bounces and d the height from which the ball was dropped.

B kdStep 2: Substitute the givens to find k. When 12 in. the ball bounces 8.4 in.d B

8.4 (12) 8.4 12

8.4 1212

k k

12k

0.7k Step 3: Substitute k into the equation from step 1. 0.7B dStep 4: Use the equation from step 3 to answer the question. If the ball bounced 56 in. how far was it dropped?

Exercise 3:

The height that a ball bounces varies directly as the height from which it was dropped. A tennis ball dropped from 12 inches bounces 8.4 inches. From what height was the tennis ball dropped if it bounces 56 inches?Step 1: Write the equation. Let B be the height the ball bounces and d the height from which the ball was dropped.

B kdStep 2: Substitute the givens to find k. When 12 in. the ball bounces 8.4 in.d B

8.4 (12) 8.4 12

8.4 1212

k k

12k

0.7k Step 3: Substitute k into the equation from step 1. 0.7B dStep 4: Use the equation from step 3 to answer the question. If the ball bounced 56 in. how far was it dropped?

In other words, if B = 56, what was d ?

Exercise 3:

The height that a ball bounces varies directly as the height from which it was dropped. A tennis ball dropped from 12 inches bounces 8.4 inches. From what height was the tennis ball dropped if it bounces 56 inches?Step 1: Write the equation. Let B be the height the ball bounces and d the height from which the ball was dropped.

B kdStep 2: Substitute the givens to find k. When 12 in. the ball bounces 8.4 in.d B

8.4 (12) 8.4 12

8.4 1212

k k

12k

0.7k Step 3: Substitute k into the equation from step 1. 0.7B dStep 4: Use the equation from step 3 to answer the question. If the ball bounced 56 in. how far was it dropped?

In other words, if B = 56, what was d ? 56 0.7d

Exercise 3:

The height that a ball bounces varies directly as the height from which it was dropped. A tennis ball dropped from 12 inches bounces 8.4 inches. From what height was the tennis ball dropped if it bounces 56 inches?Step 1: Write the equation. Let B be the height the ball bounces and d the height from which the ball was dropped.

B kdStep 2: Substitute the givens to find k. When 12 in. the ball bounces 8.4 in.d B

8.4 (12) 8.4 12

8.4 1212

k k

12k

0.7k Step 3: Substitute k into the equation from step 1. 0.7B dStep 4: Use the equation from step 3 to answer the question. If the ball bounced 56 in. how far was it dropped?

In other words, if B = 56, what was d ? 56 0.756 0.70.7 0.7

dd

Exercise 3:

The height that a ball bounces varies directly as the height from which it was dropped. A tennis ball dropped from 12 inches bounces 8.4 inches. From what height was the tennis ball dropped if it bounces 56 inches?Step 1: Write the equation. Let B be the height the ball bounces and d the height from which the ball was dropped.

B kdStep 2: Substitute the givens to find k. When 12 in. the ball bounces 8.4 in.d B

8.4 (12) 8.4 12

8.4 1212

k k

12k

0.7k Step 3: Substitute k into the equation from step 1. 0.7B dStep 4: Use the equation from step 3 to answer the question. If the ball bounced 56 in. how far was it dropped?

In other words, if B = 56, what was d ? 56 0.7

56 0.70.7

d

0.7d

Exercise 3:

The height that a ball bounces varies directly as the height from which it was dropped. A tennis ball dropped from 12 inches bounces 8.4 inches. From what height was the tennis ball dropped if it bounces 56 inches?Step 1: Write the equation. Let B be the height the ball bounces and d the height from which the ball was dropped.

B kdStep 2: Substitute the givens to find k. When 12 in. the ball bounces 8.4 in.d B

8.4 (12) 8.4 12

8.4 1212

k k

12k

0.7k Step 3: Substitute k into the equation from step 1. 0.7B dStep 4: Use the equation from step 3 to answer the question. If the ball bounced 56 in. how far was it dropped?

In other words, if B = 56, what was d ? 56 0.7

56 0.70.7

d

0.7d 80d

Exercise 3:

The height that a ball bounces varies directly as the height from which it was dropped. A tennis ball dropped from 12 inches bounces 8.4 inches. From what height was the tennis ball dropped if it bounces 56 inches?Step 1: Write the equation. Let B be the height the ball bounces and d the height from which the ball was dropped.

B kdStep 2: Substitute the givens to find k. When 12 in. the ball bounces 8.4 in.d B

8.4 (12) 8.4 12

8.4 1212

k k

12k

0.7k Step 3: Substitute k into the equation from step 1. 0.7B dStep 4: Use the equation from step 3 to answer the question. If the ball bounced 56 in. how far was it dropped?

In other words, if B = 56, what was d ? 56 0.7

56 0.70.7

d

0.7d 80d The ball was dropped

from a height of 80 inches.

Exercise 3:

The height that a ball bounces varies directly as the height from which it was dropped. A tennis ball dropped from 12 inches bounces 8.4 inches. From what height was the tennis ball dropped if it bounces 56 inches?Step 1: Write the equation. Let B be the height the ball bounces and d the height from which the ball was dropped.

B kdStep 2: Substitute the givens to find k. When 12 in. the ball bounces 8.4 in.d B

8.4 (12) 8.4 12

8.4 1212

k k

12k

0.7k Step 3: Substitute k into the equation from step 1. 0.7B dStep 4: Use the equation from step 3 to answer the question. If the ball bounced 56 in. how far was it dropped?

In other words, if B = 56, what was d ? 56 0.7

56 0.70.7

d

0.7d 80d The ball was dropped

from a height of 80 inches.

DON’T FORGET UNITS!!!

Now you try!

Now you try!Try exercise 4 on your own.

Now you try!Try exercise 4 on your own.

Exercise 4The distance required to stop a car varies directly as the square of

its speed. If 200 feet are required to stop a car traveling 60 miles per hour, how many feet are required to stop a car traveling 100 miles per hour? (round your answer to the nearest foot)

Now you try!Try exercise 4 on your own.

Exercise 4The distance required to stop a car varies directly as the square of

its speed. If 200 feet are required to stop a car traveling 60 miles per hour, how many feet are required to stop a car traveling 100 miles per hour? (round your answer to the nearest foot)

Verify that the answer is 556 feet.

Inverse Variation The relationship “y varies inversely as x” or “y is inversely proportional to x” could be written in equation form as:

where k represents the constant of proportionality

Inverse Variation The relationship “y varies inversely as x” or “y is inversely proportional to x” could be written in equation form as:

where k represents the constant of proportionality

kyx

Inverse Variation The relationship “y varies inversely as x” or “y is inversely proportional to x” could be written in equation form as:

where k represents the constant of proportionality

Exercise 5:Write a general equation that represents the following relationship:

kyx

Inverse Variation The relationship “y varies inversely as x” or “y is inversely proportional to x” could be written in equation form as:

where k represents the constant of proportionality

Exercise 5:Write a general equation that represents the following relationship:

The demand, D, for a product varies inversely as its price, P.

kyx

Inverse Variation The relationship “y varies inversely as x” or “y is inversely proportional to x” could be written in equation form as:

where k represents the constant of proportionality

Exercise 5:Write a general equation that represents the following relationship:

The demand, D, for a product varies inversely as its price, P.

kyx

kDP

Exercise 6: The water temperature, T, of the Pacific Ocean varies inversely as the water’s depth, d. At a depth of 1000 meters, the water temperature is 4.4°C. What is the water temperature at a depth of 5000 meters?

Exercise 6: The water temperature, T, of the Pacific Ocean varies inversely as the water’s depth, d. At a depth of 1000 meters, the water temperature is 4.4°C. What is the water temperature at a depth of 5000 meters?

Step 1: Write the equation.

Exercise 6: The water temperature, T, of the Pacific Ocean varies inversely as the water’s depth, d. At a depth of 1000 meters, the water temperature is 4.4°C. What is the water temperature at a depth of 5000 meters?

Step 1: Write the equation. kTd

Exercise 6: The water temperature, T, of the Pacific Ocean varies inversely as the water’s depth, d. At a depth of 1000 meters, the water temperature is 4.4°C. What is the water temperature at a depth of 5000 meters?

Step 1: Write the equation. kTd

Step 2: Substitute the givens to find k.

Exercise 6: The water temperature, T, of the Pacific Ocean varies inversely as the water’s depth, d. At a depth of 1000 meters, the water temperature is 4.4°C. What is the water temperature at a depth of 5000 meters?

Step 1: Write the equation. kTd

Step 2: Substitute the givens to find k. When 1000, 4.4 C d T

Exercise 6: The water temperature, T, of the Pacific Ocean varies inversely as the water’s depth, d. At a depth of 1000 meters, the water temperature is 4.4°C. What is the water temperature at a depth of 5000 meters?

Step 1: Write the equation. kTd

Step 2: Substitute the givens to find k. When 1000, 4.4 C d T

4.41000k

Exercise 6: The water temperature, T, of the Pacific Ocean varies inversely as the water’s depth, d. At a depth of 1000 meters, the water temperature is 4.4°C. What is the water temperature at a depth of 5000 meters?

Step 1: Write the equation. kTd

Step 2: Substitute the givens to find k. When 1000, 4.4 C d T

4.41000

4.4(1000) (1000)1000

k

k

Exercise 6: The water temperature, T, of the Pacific Ocean varies inversely as the water’s depth, d. At a depth of 1000 meters, the water temperature is 4.4°C. What is the water temperature at a depth of 5000 meters?

Step 1: Write the equation. kTd

Step 2: Substitute the givens to find k. When 1000, 4.4 C d T

4.41000

4.4(1000)1000

k

k

(1000

)

Exercise 6: The water temperature, T, of the Pacific Ocean varies inversely as the water’s depth, d. At a depth of 1000 meters, the water temperature is 4.4°C. What is the water temperature at a depth of 5000 meters?

Step 1: Write the equation. kTd

Step 2: Substitute the givens to find k. When 1000, 4.4 C d T

4.41000

4.4(1000)1000

k

k

(1000

)

4400k

Exercise 6: The water temperature, T, of the Pacific Ocean varies inversely as the water’s depth, d. At a depth of 1000 meters, the water temperature is 4.4°C. What is the water temperature at a depth of 5000 meters?

Step 1: Write the equation. kTd

Step 2: Substitute the givens to find k. When 1000, 4.4 C d T

4.41000

4.4(1000) (1000)1000

4400

k

k

k

Step 3: Substitute k into the equation from step 1.

Exercise 6: The water temperature, T, of the Pacific Ocean varies inversely as the water’s depth, d. At a depth of 1000 meters, the water temperature is 4.4°C. What is the water temperature at a depth of 5000 meters?

Step 1: Write the equation. kTd

Step 2: Substitute the givens to find k. When 1000, 4.4 C d T

4.41000

4.4(1000) (1000)1000

4400

k

k

k

Step 3: Substitute k into the equation from step 1.

4400Td

Exercise 6: The water temperature, T, of the Pacific Ocean varies inversely as the water’s depth, d. At a depth of 1000 meters, the water temperature is 4.4°C. What is the water temperature at a depth of 5000 meters?

Step 1: Write the equation. kTd

Step 2: Substitute the givens to find k. When 1000, 4.4 C d T

4.41000

4.4(1000) (1000)1000

4400

k

k

k

Step 3: Substitute k into the equation from step 1.

Step 4: Use the equation from step 3 to answer the question.

4400Td

Exercise 6: The water temperature, T, of the Pacific Ocean varies inversely as the water’s depth, d. At a depth of 1000 meters, the water temperature is 4.4°C. What is the water temperature at a depth of 5000 meters?

Step 1: Write the equation. kTd

Step 2: Substitute the givens to find k. When 1000, 4.4 C d T

4.41000

4.4(1000) (1000)1000

4400

k

k

k

Step 3: Substitute k into the equation from step 1.

Step 4: Use the equation from step 3 to answer the question. What is the water temperature at a depth of 5000 meters?

4400Td

Exercise 6: The water temperature, T, of the Pacific Ocean varies inversely as the water’s depth, d. At a depth of 1000 meters, the water temperature is 4.4°C. What is the water temperature at a depth of 5000 meters?

Step 1: Write the equation. kTd

Step 2: Substitute the givens to find k. When 1000, 4.4 C d T

4.41000

4.4(1000) (1000)1000

4400

k

k

k

Step 3: Substitute k into the equation from step 1.

Step 4: Use the equation from step 3 to answer the question. What is the water temperature at a depth of 5000 meters?

In other words, if d = 1000, what is T ?

4400Td

Exercise 6: The water temperature, T, of the Pacific Ocean varies inversely as the water’s depth, d. At a depth of 1000 meters, the water temperature is 4.4°C. What is the water temperature at a depth of 5000 meters?

Step 1: Write the equation. kTd

Step 2: Substitute the givens to find k. When 1000, 4.4 C d T

4.41000

4.4(1000) (1000)1000

4400

k

k

k

Step 3: Substitute k into the equation from step 1.

Step 4: Use the equation from step 3 to answer the question. What is the water temperature at a depth of 5000 meters?

In other words, if d = 1000, what is T ?

4400Td

44005000

T

Exercise 6: The water temperature, T, of the Pacific Ocean varies inversely as the water’s depth, d. At a depth of 1000 meters, the water temperature is 4.4°C. What is the water temperature at a depth of 5000 meters?

Step 1: Write the equation. kTd

Step 2: Substitute the givens to find k. When 1000, 4.4 C d T

4.41000

4.4(1000) (1000)1000

4400

k

k

k

Step 3: Substitute k into the equation from step 1.

Step 4: Use the equation from step 3 to answer the question. What is the water temperature at a depth of 5000 meters?

In other words, if d = 1000, what is T ?

4400Td

4400 0.885000

T C

Exercise 6: The water temperature, T, of the Pacific Ocean varies inversely as the water’s depth, d. At a depth of 1000 meters, the water temperature is 4.4°C. What is the water temperature at a depth of 5000 meters?

Step 1: Write the equation. kTd

Step 2: Substitute the givens to find k. When 1000, 4.4 C d T

4.41000

4.4(1000) (1000)1000

4400

k

k

k

Step 3: Substitute k into the equation from step 1.

Step 4: Use the equation from step 3 to answer the question. What is the water temperature at a depth of 5000 meters?

In other words, if d = 1000, what is T ?

The temperature is 0.88°C at a depth of 5000 m.

4400Td

4400 0.885000

T C

Inverse Variation intuitively…

Inverse Variation intuitively…

Notice what happened in the last example:

Inverse Variation intuitively…

Notice what happened in the last example:

When the depth got bigger, the temperature got smaller.

Inverse Variation intuitively…

Notice what happened in the last example:

When the depth got bigger, the temperature got smaller.

This makes sense because the farther underwater you go in the ocean the colder the temperature gets.

Inverse Variation intuitively…

Notice what happened in the last example:

When the depth got bigger, the temperature got smaller.

This makes sense because the farther underwater you go in the ocean the colder the temperature gets.

In general, for inverse variation…

Inverse Variation intuitively…

Notice what happened in the last example:

When the depth got bigger, the temperature got smaller.

This makes sense because the farther underwater you go in the ocean the colder the temperature gets.

In general, for inverse variation…

• When one variable increases the other decreases.

Inverse Variation intuitively…

Notice what happened in the last example:

When the depth got bigger, the temperature got smaller.

This makes sense because the farther underwater you go in the ocean the colder the temperature gets.

In general, for inverse variation…

• When one variable increases the other decreases.

• When one variable decreases the other increases.

Inverse Variation intuitively…

Notice what happened in the last example:

When the depth got bigger, the temperature got smaller.

This makes sense because the farther underwater you go in the ocean the colder the temperature gets.

In general, for inverse variation…

• When one variable increases the other decreases.

• When one variable decreases the other increases.

• Basically they do the opposite….that is why we call it inverse variation.

Combined VariationIn combined variation, direct and inverse variation occur at the same

time.

Combined VariationIn combined variation, direct and inverse variation occur at the same

time.

Example:

The sale of a product varies directly as its advertising budget and inversely as the price of the product.

Combined VariationIn combined variation, direct and inverse variation occur at the same

time.

Example:

The sale of a product varies directly as its advertising budget and inversely as the price of the product.

We could write this relationship as

kASP

Combined VariationIn combined variation, direct and inverse variation occur at the same

time.

Example:

The sale of a product varies directly as its advertising budget and inversely as the price of the product.

We could write this relationship as

Notice that since sales vary directly with advertising then if we increased our advertising budget we would expect sales to also increase.

kASP

Combined VariationIn combined variation, direct and inverse variation occur at the same time.

Example:

The sale of a product varies directly as its advertising budget and inversely as the price of the product.

We could write this relationship as

Notice that since sales vary directly with advertising then if we increased our advertising budget we would expect sales to also increase.

Notice that since sales vary inversely with price then if we increased the price of the product we would expect sales to decrease.

kASP

Exercise 7: The number of minutes needed to finish a math assignment varies directly as the number of problems and inversely as the number of people working to solve the problems. If it takes 4 people 32 minutes to solve 16 problems, how many minutes will it take 8 people to solve 24 problems?

Exercise 7: The number of minutes needed to finish a math assignment varies directly as the number of problems and inversely as the number of people working to solve the problems. If it takes 4 people 32 minutes to solve 16 problems, how many minutes will it take 8 people to solve 24 problems?Step 1: Write the equation.

Exercise 7: The number of minutes needed to finish a math assignment varies directly as the number of problems and inversely as the number of people working to solve the problems. If it takes 4 people 32 minutes to solve 16 problems, how many minutes will it take 8 people to solve 24 problems?Step 1: Write the equation. Let m = number of minutes, n = number of problems, and p = number of people

Exercise 7: The number of minutes needed to finish a math assignment varies directly as the number of problems and inversely as the number of people working to solve the problems. If it takes 4 people 32 minutes to solve 16 problems, how many minutes will it take 8 people to solve 24 problems?Step 1: Write the equation. Let m = number of minutes, n = number of problems, and p = number of people knm

p

Exercise 7: The number of minutes needed to finish a math assignment varies directly as the number of problems and inversely as the number of people working to solve the problems. If it takes 4 people 32 minutes to solve 16 problems, how many minutes will it take 8 people to solve 24 problems?Step 1: Write the equation. Let m = number of minutes, n = number of problems, and p = number of people knm

p

Step 2: Substitute the givens to find k.

Exercise 7: The number of minutes needed to finish a math assignment varies directly as the number of problems and inversely as the number of people working to solve the problems. If it takes 4 people 32 minutes to solve 16 problems, how many minutes will it take 8 people to solve 24 problems?Step 1: Write the equation. Let m = number of minutes, n = number of problems, and p = number of people knm

p

Step 2: Substitute the givens to find k. When 4 people, 32 minutes and 16 problems p m n

Exercise 7: The number of minutes needed to finish a math assignment varies directly as the number of problems and inversely as the number of people working to solve the problems. If it takes 4 people 32 minutes to solve 16 problems, how many minutes will it take 8 people to solve 24 problems?Step 1: Write the equation. Let m = number of minutes, n = number of problems, and p = number of people knm

p

Step 2: Substitute the givens to find k. When 4 people, 32 minutes and 16 problems p m n

(16)324

k

Exercise 7: The number of minutes needed to finish a math assignment varies directly as the number of problems and inversely as the number of people working to solve the problems. If it takes 4 people 32 minutes to solve 16 problems, how many minutes will it take 8 people to solve 24 problems?Step 1: Write the equation. Let m = number of minutes, n = number of problems, and p = number of people knm

p

Step 2: Substitute the givens to find k. When 4 people, 32 minutes and 16 problems p m n

(16) 1632 324 4

k k

Exercise 7: The number of minutes needed to finish a math assignment varies directly as the number of problems and inversely as the number of people working to solve the problems. If it takes 4 people 32 minutes to solve 16 problems, how many minutes will it take 8 people to solve 24 problems?Step 1: Write the equation. Let m = number of minutes, n = number of problems, and p = number of people knm

p

Step 2: Substitute the givens to find k. When 4 people, 32 minutes and 16 problems p m n

(16) 1632 32 32 44 4

k k k

Exercise 7: The number of minutes needed to finish a math assignment varies directly as the number of problems and inversely as the number of people working to solve the problems. If it takes 4 people 32 minutes to solve 16 problems, how many minutes will it take 8 people to solve 24 problems?Step 1: Write the equation. Let m = number of minutes, n = number of problems, and p = number of people knm

p

Step 2: Substitute the givens to find k. When 4 people, 32 minutes and 16 problems p m n

(16) 1632 32 32 4 84 4

k k k k

Exercise 7: The number of minutes needed to finish a math assignment varies directly as the number of problems and inversely as the number of people working to solve the problems. If it takes 4 people 32 minutes to solve 16 problems, how many minutes will it take 8 people to solve 24 problems?Step 1: Write the equation. Let m = number of minutes, n = number of problems, and p = number of people knm

p

Step 2: Substitute the givens to find k. When 4 people, 32 minutes and 16 problems p m n

(16) 1632 32 32 4 84 4

k k k k

Step 3: Substitute k into the equation from step 1.

Exercise 7: The number of minutes needed to finish a math assignment varies directly as the number of problems and inversely as the number of people working to solve the problems. If it takes 4 people 32 minutes to solve 16 problems, how many minutes will it take 8 people to solve 24 problems?Step 1: Write the equation. Let m = number of minutes, n = number of problems, and p = number of people knm

p

Step 2: Substitute the givens to find k. When 4 people, 32 minutes and 16 problems p m n

(16) 1632 32 32 4 84 4

k k k k

Step 3: Substitute k into the equation from step 1. 8nmp

Exercise 7: The number of minutes needed to finish a math assignment varies directly as the number of problems and inversely as the number of people working to solve the problems. If it takes 4 people 32 minutes to solve 16 problems, how many minutes will it take 8 people to solve 24 problems?Step 1: Write the equation. Let m = number of minutes, n = number of problems, and p = number of people knm

p

Step 2: Substitute the givens to find k. When 4 people, 32 minutes and 16 problems p m n

(16) 1632 32 32 4 84 4

k k k k

Step 3: Substitute k into the equation from step 1.

Step 4: Use the equation from step 3 to answer the question.

8nmp

Exercise 7: The number of minutes needed to finish a math assignment varies directly as the number of problems and inversely as the number of people working to solve the problems. If it takes 4 people 32 minutes to solve 16 problems, how many minutes will it take 8 people to solve 24 problems?Step 1: Write the equation. Let m = number of minutes, n = number of problems, and p = number of people knm

p

Step 2: Substitute the givens to find k. When 4 people, 32 minutes and 16 problems p m n

(16) 1632 32 32 4 84 4

k k k k

Step 3: Substitute k into the equation from step 1.

Step 4: Use the equation from step 3 to answer the question. How many minutes will it take 8 people to solve 24 problems?

8nmp

Exercise 7: The number of minutes needed to finish a math assignment varies directly as the number of problems and inversely as the number of people working to solve the problems. If it takes 4 people 32 minutes to solve 16 problems, how many minutes will it take 8 people to solve 24 problems?Step 1: Write the equation. Let m = number of minutes, n = number of problems, and p = number of people knm

p

Step 2: Substitute the givens to find k. When 4 people, 32 minutes and 16 problems p m n

(16) 1632 32 32 4 84 4

k k k k

Step 3: Substitute k into the equation from step 1.

Step 4: Use the equation from step 3 to answer the question. How many minutes will it take 8 people to solve 24 problems?

In other words, if p = 8 and n = 24, what is m ?

8nmp

Exercise 7: The number of minutes needed to finish a math assignment varies directly as the number of problems and inversely as the number of people working to solve the problems. If it takes 4 people 32 minutes to solve 16 problems, how many minutes will it take 8 people to solve 24 problems?Step 1: Write the equation. Let m = number of minutes, n = number of problems, and p = number of people knm

p

Step 2: Substitute the givens to find k. When 4 people, 32 minutes and 16 problems p m n

(16) 1632 32 32 4 84 4

k k k k

Step 3: Substitute k into the equation from step 1.

Step 4: Use the equation from step 3 to answer the question. How many minutes will it take 8 people to solve 24 problems?

In other words, if p = 8 and n = 24, what is m ? 8(24)

8m

8nmp

Exercise 7: The number of minutes needed to finish a math assignment varies directly as the number of problems and inversely as the number of people working to solve the problems. If it takes 4 people 32 minutes to solve 16 problems, how many minutes will it take 8 people to solve 24 problems?Step 1: Write the equation. Let m = number of minutes, n = number of problems, and p = number of people knm

p

Step 2: Substitute the givens to find k. When 4 people, 32 minutes and 16 problems p m n

(16) 1632 32 32 4 84 4

k k k k

Step 3: Substitute k into the equation from step 1.

Step 4: Use the equation from step 3 to answer the question. How many minutes will it take 8 people to solve 24 problems?

In other words, if p = 8 and n = 24, what is m ? 8m

(24)8

24

8nmp

Exercise 7: The number of minutes needed to finish a math assignment varies directly as the number of problems and inversely as the number of people working to solve the problems. If it takes 4 people 32 minutes to solve 16 problems, how many minutes will it take 8 people to solve 24 problems?Step 1: Write the equation. Let m = number of minutes, n = number of problems, and p = number of people knm

p

Step 2: Substitute the givens to find k. When 4 people, 32 minutes and 16 problems p m n

(16) 1632 32 32 4 84 4

k k k k

Step 3: Substitute k into the equation from step 1.

Step 4: Use the equation from step 3 to answer the question. How many minutes will it take 8 people to solve 24 problems?

In other words, if p = 8 and n = 24, what is m ?

It will take 24 minutes for 8 people to solve 24 problems.

8m(24)8

24

8nmp

Joint Variation The relationship “y varies jointly as x and z” could be written in equation form as:

where k represents the constant of proportionality

y kxz

Exercise 8: The Volume, V, of a cone varies jointly as the area, B, of the base and the height, h, of the cone. If V = 15.7 in.3 when B = 9.42 in. and h = 5 in., what is the volume of a cone with B = 20 in. and h = 8 in ?

Exercise 8: The Volume, V, of a cone varies jointly as the area, B, of the base and the height, h, of the cone. If V = 15.7 in.3 when B = 9.42 in. and h = 5 in., what is the volume of a cone with B = 20 in. and h = 8 in ?

Step 1: Write the equation.

Exercise 8: The Volume, V, of a cone varies jointly as the area, B, of the base and the height, h, of the cone. If V = 15.7 in.3 when B = 9.42 in. and h = 5 in., what is the volume of a cone with B = 20 in. and h = 8 in ?

Step 1: Write the equation. V kBh

Exercise 8: The Volume, V, of a cone varies jointly as the area, B, of the base and the height, h, of the cone. If V = 15.7 in.3 when B = 9.42 in. and h = 5 in., what is the volume of a cone with B = 20 in. and h = 8 in ?

Step 1: Write the equation. V kBh

Step 2: Substitute the givens to find k.

Exercise 8: The Volume, V, of a cone varies jointly as the area, B, of the base and the height, h, of the cone. If V = 15.7 in.3 when B = 9.42 in. and h = 5 in., what is the volume of a cone with B = 20 in. and h = 8 in ?

Step 1: Write the equation. V kBh

Step 2: Substitute the givens to find k. When 15.7, 9.42, and 5. V B h

Exercise 8: The Volume, V, of a cone varies jointly as the area, B, of the base and the height, h, of the cone. If V = 15.7 in.3 when B = 9.42 in. and h = 5 in., what is the volume of a cone with B = 20 in. and h = 8 in ?

Step 1: Write the equation. V kBh

Step 2: Substitute the givens to find k. When 15.7, 9.42, and 5. V B h

15.7 (9.42)(5)k

Exercise 8: The Volume, V, of a cone varies jointly as the area, B, of the base and the height, h, of the cone. If V = 15.7 in.3 when B = 9.42 in. and h = 5 in., what is the volume of a cone with B = 20 in. and h = 8 in ?

Step 1: Write the equation. V kBh

Step 2: Substitute the givens to find k. When 15.7, 9.42, and 5. V B h

15.7 (9.42)(5) 15.7 47.1k k

Exercise 8: The Volume, V, of a cone varies jointly as the area, B, of the base and the height, h, of the cone. If V = 15.7 in.3 when B = 9.42 in. and h = 5 in., what is the volume of a cone with B = 20 in. and h = 8 in ?

Step 1: Write the equation. V kBh

Step 2: Substitute the givens to find k. When 15.7, 9.42, and 5. V B h 15.7 47.115.7 (9.42)(5) 15.7 47.147.1 47.1

kk k

Exercise 8: The Volume, V, of a cone varies jointly as the area, B, of the base and the height, h, of the cone. If V = 15.7 in.3 when B = 9.42 in. and h = 5 in., what is the volume of a cone with B = 20 in. and h = 8 in ?

Step 1: Write the equation. V kBh

Step 2: Substitute the givens to find k. When 15.7, 9.42, and 5. V B h 15.7 47.115.7 (9.42)(5) 15.7 47.147.1

k k 47.1k

Exercise 8: The Volume, V, of a cone varies jointly as the area, B, of the base and the height, h, of the cone. If V = 15.7 in.3 when B = 9.42 in. and h = 5 in., what is the volume of a cone with B = 20 in. and h = 8 in ?

Step 1: Write the equation. V kBh

Step 2: Substitute the givens to find k. When 15.7, 9.42, and 5. V B h 15.7 47.115.7 (9.42)(5) 15.7 47.147.1

k k 47.1k

10.3333333

k

Exercise 8: The Volume, V, of a cone varies jointly as the area, B, of the base and the height, h, of the cone. If V = 15.7 in.3 when B = 9.42 in. and h = 5 in., what is the volume of a cone with B = 20 in. and h = 8 in ?

Step 1: Write the equation. V kBh

Step 2: Substitute the givens to find k. When 15.7, 9.42, and 5. V B h

Step 3: Substitute k into the equation from step 1.

15.7 47.115.7 (9.42)(5) 15.7 47.147.1

k k 47.1k

10.3333333

k

Exercise 8: The Volume, V, of a cone varies jointly as the area, B, of the base and the height, h, of the cone. If V = 15.7 in.3 when B = 9.42 in. and h = 5 in., what is the volume of a cone with B = 20 in. and h = 8 in ?

Step 1: Write the equation. V kBh

Step 2: Substitute the givens to find k. When 15.7, 9.42, and 5. V B h

Step 3: Substitute k into the equation from step 1. 13

V Bh

15.7 47.115.7 (9.42)(5) 15.7 47.147.1

k k 47.1k

10.3333333

k

Exercise 8: The Volume, V, of a cone varies jointly as the area, B, of the base and the height, h, of the cone. If V = 15.7 in.3 when B = 9.42 in. and h = 5 in., what is the volume of a cone with B = 20 in. and h = 8 in ?

Step 1: Write the equation. V kBh

Step 2: Substitute the givens to find k. When 15.7, 9.42, and 5. V B h

Step 3: Substitute k into the equation from step 1.

Step 4: Use the equation from step 3 to answer the question.

13

V Bh

15.7 47.115.7 (9.42)(5) 15.7 47.147.1

k k 47.1k

10.3333333

k

Exercise 8: The Volume, V, of a cone varies jointly as the area, B, of the base and the height, h, of the cone. If V = 15.7 in.3 when B = 9.42 in. and h = 5 in., what is the volume of a cone with B = 20 in. and h = 8 in ?

Step 1: Write the equation. V kBh

Step 2: Substitute the givens to find k. When 15.7, 9.42, and 5. V B h

Step 3: Substitute k into the equation from step 1.

Step 4: Use the equation from step 3 to answer the question. What is the volume of a cone with B = 20 in. and h = 8 in. ?

13

V Bh

15.7 47.115.7 (9.42)(5) 15.7 47.147.1

k k 47.1k

10.3333333

k

Exercise 8: The Volume, V, of a cone varies jointly as the area, B, of the base and the height, h, of the cone. If V = 15.7 in.3 when B = 9.42 in. and h = 5 in., what is the volume of a cone with B = 20 in. and h = 8 in ?

Step 1: Write the equation. V kBh

Step 2: Substitute the givens to find k. When 15.7, 9.42, and 5. V B h

Step 3: Substitute k into the equation from step 1.

Step 4: Use the equation from step 3 to answer the question. What is the volume of a cone with B = 20 in. and h = 8 in. ?

31 160(20)(8) 53.33in.3 3

V

13

V Bh

15.7 47.115.7 (9.42)(5) 15.7 47.147.1

k k 47.1k

10.3333333

k

Exercise 8: The Volume, V, of a cone varies jointly as the area, B, of the base and the height, h, of the cone. If V = 15.7 in.3 when B = 9.42 in. and h = 5 in., what is the volume of a cone with B = 20 in. and h = 8 in ?

Step 1: Write the equation. V kBh

Step 2: Substitute the givens to find k. When 15.7, 9.42, and 5. V B h

Step 3: Substitute k into the equation from step 1.

Step 4: Use the equation from step 3 to answer the question. What is the volume of a cone with B = 20 in. and h = 8 in. ?

The volume of a cone with B = 20 in. and h = 8 in. is approximately 53.33 cubic inches.

31 160(20)(8) 53.33in.3 3

V

13

V Bh

15.7 47.115.7 (9.42)(5) 15.7 47.147.1

k k 47.1k

10.3333333

k