Definition:When two or more related variables are changing with respect to time they are called related rates

Section 2-6 Related Rates

Derivative = rate of change

• When we say “rate of change” without specifying what it's with respect to, we mean “rate of change with respect to time”.

– example: velocity- change in distance with respect to time

• If the variable is different from time use implicit differentiation

• Unless otherwise specified assume rate of change means with respect to time

1) Differentiate each with respect to t.

2rA

3

3

4rV

222 bac

50tan

y

2) Suppose x and y are both differentiable functions with respect to time, t, and are related by the equation y = x3 + 5 find dy/dt when x = 3, given that dx/dt = 2 when x = 3.

~ Differentiate both sides of the equation with respect to t using the chain rule.

3) Suppose x and y are both differentiable functions with respect to time, t, and are related by the equation x2y = 2. Find dy/dt when x = 3, given that dx/dt = 2 when x = 3.

Helpful suggestions for solving Related Rates

• Identify all known and unknown quantities. Make a sketch and label it.

• Find an equation that describes the relationship between the variables. Common equations include the Pythagorean Theorem, Area of circle, Volumes of known geometric shape, SOH-CAH-TOA, similar triangles, etc.

• Differentiate both sides of the equation with respect to time.

• Substitute all known quantities and rates into the equation and solve.

4) A 13 – foot ladder is leaning against the wall of a house. The base of the ladder slides away from the wall at a rate of 0.75 feet per second. How fast is the top of the ladder moving down the wall when the base is 12 feet from the wall?

5) Oil spills into a lake in a circular pattern. If the radius of the circle increases at a constant rate of 3 feet per second, how fast is the area of the spill increasing at the end of 30 minutes.

6) A baseball diamond is a square with side 90 feet. A batter hits the ball and runs toward first base with a speed of 24 feet per second. At what rate is his distance from second base decreasing when he is halfway to first base?

7) A weather balloon is released 50 feet from an observer. It rises at a rate of 8 feet per second. How fast is the angle of elevation changing when the balloon is 50 feet high?

Ɵ

8) A man 6 feet tall is walking toward a lamppost 20 feet high at a rate of 5 feet per second.  The light at the top of the lamppost (20 feet above the ground) is casting a shadow of the man.  At what rate is the tip of his shadow moving and at what rate is the length of his shadow changing when he is 10 feet from the base of the lamppost? 

9) Two cars start moving from the same point at noon. One travels south at 60 mi/h and the other travels west at 25 mi/h. At what rate is the distance between the cars increasing at 2:00 pm?

10) Water is being poured into a conical reservoir at the rate of pi cubic feet per second.  The reservoir has a radius of 6 feet across the top and a height of 12 feet.  At what rate is the depth of the water increasing when the depth is 6 feet?

11) Air is being pumped into a spherical ball at a rate of 5 cubic centimeters per minute. Determine the rate at which the radius of the ball is increasing when the diameter of the ball is 20 cm.

12) A 15 foot ladder is resting against the wall. The bottom is initially 10 feet away from the wall and is being pushed towards the wall at a rate of .25 ft/sec. How fast is the top of the ladder moving up the wall 12 seconds after we start pushing?

15 ft

13) A family and a guy are 50 feet apart. The family starts walking north at a rate so that the angle between them is changing at a constant rate of 0.01 radians per minute. At what rate is the distance between them changing when radians?

5.0

14) Two people on bikes are separated by 350 meters. Person A starts riding north at a rate of 5m/sec and 7 minutes later Person B starts riding south at 3m/sec. At what rate is the distance separating the two people changing 25 minutes after person A starts riding?

A B

Homework

Page 154

Problems # 1, 3,13,17, 20, 25, 29, 31, 33, and 43