dmat 0093 notes

DMAT 0093

Write a quadratic equation in standard form that has two solutions, 10 and -4?

If 10 and -4 are solutions, the equation will look like this:

(x - 10)(x+4) = 0

If you distribute you will end up with:

x^2 - 6x - 40 = 0

In Mathway – “Find the Standard Form of the Equation”

Write a quadratic equation in standard form that has two solutions, 5 and -3?

(x-5)(x+3)=0

Answer will show x^2-2x=15 but show it as x^2-2x-15 in answer

PERIMETER is distance around therefore you ADD the figures.

Represent the given condition using a single variable, x.

The length and width of a rectangle whose length is 7 centimeters more than its width.

x

width = x

length = x+7


Three Consecutive integers.

X is the first or smallest integer. The expression for the next consecutive integer in terms of x is one greater than x.

X+1

The third consecutive integer would be one greater than the integer

X+1+1=x+2

Therefore, three consecutive integers where the variable x is an integer are x, x+1 and x+2


The base and height of a triangle whose height is two more than three times its base.

X

If three times the base is 3x then “two more” than three times the base is 3x+2

Base = x

Height = 3x+2

The area of the square is 49 square units. Find the length of its sides.

49=x^2

x=7,-7 (Since the length of a side can’t be negative -7, the answer is 7.) The length of a side is 7.

The area of the square is 100 square units. Find the length of its sides.

100=x^2

x=10,-10 (Since the length of a side can’t be negative -10, the answer is 10.) The length of a side is 10

The perimeter of the quadrilateral is 45 centimeters. Find the length of the side.

X+7

X+6 x^2-3x

3x-3

Solve for X:

45=(x+6)+(x+7)+(x^2-3x)+(3x-3)

X=-7,5 Since the sides can’t be negative, it has to be 5

The perimeter of the quadrilateral is 52 centimeters. Find the length of the side.

X+4

X+3 x^2-3x

3x-3

Solve for X:

52=(x+3)+(x+4)+(x^2-3x)+(3x-3)

X=-8,6 Since the sides can’t be negative, it has to be 6.

An object is thrown upward from the top of a 208 foot building with an initial velocity of 192 feet per second. The height h of the object after t seconds is given by the quadratic equation h=-16t^2+192t+208. When will the object hit the ground?

To find out how long it takes the object to hit the ground, find the value of t when the height is h=0.

0=-16t^2+192t+208

T=13,-1 Since the time t cannot be negative, the answer is 13

Rewrite the rational expression as an equivalent rational expression with the given denominator.

ANSWER:

Rewrite the rational expression with the given denominator:

In mathway, “solve the equation for x”

Add and Simplify:

In mathway, use “combine the expressions"

ANSWER:

Perform the indicated operation:

In mathway, use “combine the expressions"

The length of a rectangle is 5 centimeters less than seven times its width. Its area is 18 square centimeters. Find the dimensions.

7x-5

Width = x

Length =7x-5

Area = length * width

18 = 7x-5 * x

Write in standard form: 18=x(7x-5)

X=-9/7, 2 Since it can’t be negative, 2 is the answer

Example:

The length of a rectangle is 27 centimeters less than five times its width. Its area is 56 square centimeters. Find the dimensions.

5x-27

56=5x-27*x or 56=x(5x-27)

X=-8/5,7 Since it can’t be negative, 7 is the answer

The width is 7 cm

The length is 5x-27 or 5(7)-27=8 cm

The equation D=1/2n(n-3) gives the number of diagonals D for a polygon with n sides. Use this equation to find the number of diagonals for a polygon with 20 sides.

Substitute 20 for n into the given formula:

D=1/2n(n-3)

D =1/2 20(20-3)

D =170

Therefore, the polygon with 20 sides has 170 diagonals.

EXAMPLE: The equation D=1/2n(n-3) gives the number of diagonals D for a polygon with n sides. Use this equation to find the number of diagonals for a polygon with 9 sides.

Substitute 9 for n into the given formula:

D=1/2n(n-3)

D =1/2 9(9-3)

D =27

The sum of a number and its square is 156. Find the number.

X+x^2=156 written in standard form: x^2+x-156=0

x=-13,12

EXAMPLE:

The sum of a number and its square is 2. Find the number.

X+x^2=2 written in standard form: x^2+x-2=0

x=-2,1

If the sides of a square are increased by 6 inches, the area becomes 225 square inches. Find the length of the sides of the original square.

X x+6

Area = side^2

225 = (x+6)^2

Written in standard form:

225=(x+6)^2

x=9,-21 since length of side can’t be negative, it’s 9 in.

EXAMPLE:

If the sides of a square are increased by 2 inches, the area becomes 36 square inches. Find the length of the sides of the original square.

X x+2

Area = side^2

36 = (x+2)^2

Written in standard form:

36=(x+2)^2

x=4,-8 since length of side can’t be negative, it’s 4 in.

The ratio of the weight of an object on Planet A to the weight of an object on Planet B is 100 to 3. If an elephant weighs 5000 pounds on Planet A, find the elephant’s weight on Planet B.

Let w represent the weight of the elephant on Planet B. Translate into an equation: 100 is to 3 as 5000 is to w.

Weight on Planet A = elephants weight on Planet AWeight on Planet B elephants weight on Planet B

100 = 50003 wW=150

There are 110 calories per 28.4 grams of Cereal X. Find how many calories are in 45.44 grams of this cereal.

Original number of calories = new number of caloriesOriginal number of grams new number of grams

110 = c 28.4 45.44C=176

There are 110 calories per 28.4 grams of Cereal X. Find how many calories are in 31.24 grams of this cereal.

Original number of calories = new number of caloriesOriginal number of grams new number of grams

110 = c 28.4 31.24C=121

Find the unknown length y in the following pair of similar triangles.

31 39 7.75 y

Large triangle leg = large triangle hypotenuseSmall triangle leg small triangle hypotenuse

31 = 39 7.75 yY=9.75

Find the unknown length y in the following pair of similar triangles.

13 17 3.25 y

Large triangle leg = large triangle hypotenuseSmall triangle leg small triangle hypotenuse

13 = 173.25 yY=4.25

A human factors expert recommends that there be at least 9 sq ft of floor space in a college classroom for every student in the class. Find the minimum floor space that 86 students require.

Square ft = square feetStudents students

9 = n 1 86N=774

Tom Moss, a tailback, holds a high school sports record for total yards rushed in a season. In 1953, he rushed for 5419 total yards in 16 games. Find his average rushing yards per game.

1 game = yards per 1 game12 games yards per 12 games

1 = x16 5419X=who knows

The manufacturers of cans of salted mixed nuts state that the ratio of peanuts to other nuts is 7 to 5. If 462 peanuts are in a can, find how many other nuts should also be in the can.

7 = 4625 xX=330

The manufacturers of cans of salted mixed nuts state that the ratio of peanuts to other nuts is 4 to 3. If 316 peanuts are in a can, find how many other nuts should also be in the can.

4 = 3163 x

X=237

In 2 minutes, a conveyor belt moves 700 pounds of recyclable aluminum from the delivery truck to a storage area. A smaller belt moves the same quantity of cans the same distance in 4 minutes. If both belts are used, find how long it takes to move the cans to the storage area.

1 + 1 = 12 4 xX=4/3 simplify that and x=1 1/3


1 + 1 = 16 9 xX=18/5 simplify that and x=3 3/5

One custodian cleans a suite of offices in 5 hours. When a second worker is asked to join the regular custodian, the job takes only 2 ½ hours. How long does it take the second worker to do the same job alone?

One custodian @ 5 hrs two custodians @ 5 hours

1 + 1 = 25 x 5X=5

One custodian cleans a suite of offices in 3 hours. When a second worker is asked to join the regular custodian, the job takes only 1 ½ hours. How long does it take the second worker to do the same job alone?

One custodian @ 3 hrs two custodians @ 3 hours

1 + 1 = 23 x 3X=3

One pump fills a tank 3 times as fast as another pump. If the pump works together, they fill the tank in 27 minutes. How long does it take for each pump to fill the tank?

One pump three times faster

1 + 1 = 1x 3x 27X= 36 then times 3 (for three times faster)=108

One pump fills a tank 2 times as fast as another pump. If the pump works together, they fill the tank in 16 minutes. How long does it take for each pump to fill the tank?

One pump three times faster

1 + 1 = 1x 2x 16X= 24 then times 2 (for two times faster)=48

It takes the faster pump 24 minutes to fill the tank when working alone and it takes the slower pump 48 minutes.


1 + 1 = 16 9 xX= 18/5 – simplify for 3 3/5 minutes

The conveyor belts can move the 500 pounds of recyclable aluminum from the delivery truck to a storage area in 3 3/5minutes.


1 + 1 = 15 8 xX= 40/13 simplify 3 1/13

The conveyor belts can move the 400 pounds of recyclable aluminum from the delivery truck to a storage area in 3 1/13 minutes.

Smith engineering is in the process of reviewing the salaries of their surveyors. During this review the company found that and experience surveyor can survey a roadbed in 3 hours. An apprentice surveyor needs 7 hours to survey the same stretch of road. If the two worked together find how long it takes them to complete the job.-------------------------Experience surveyor rate : 1/3 job/hrApprentice rate: 1/7 job/hrTogether rate: 1/x job/hr--------------------------------Equation:rate + rate + together rate:1/3 + 1/7 = 1/x7x + 3x = 2110x = 21x = 2.1 hours (time it will take if they work together)

Use function notation to write the equation of the line with the given slope and y-intercept.

Slope -6; y-intercept (0,11)

ANSWER: f(x)=-6x+11


Slope -7; y-intercept (0,9)

ANSWER: f(x)=-7x+9


Slope 4; y-intercept (0,-7/9)

ANSWER: f(x)=4x+(-7/9)


Slope 5/7; y-intercept (0,0)

ANSWER: f(x)=5/7x

Find an equation of the line with the given slope and containing the given point.

WEBMATH: http://www.webmath.com/equline2.html

Find an equation of the line passing through the given points.


Write an equation of the line using function notation.

Horizontal; through (-2,-1)

ANSWER: f(x)=-1

Write an equation of the line using function notation.

Horizontal; through (0,-1)

ANSWER: f(x)=-1

Find an equation of the line through (1,1) and parallel to y=5x+3.


Enter as (1,1) and slope is 5 (find slope at mathway entering y=5x+3)

ANSWER : f(x)=5x-4

Find an equation of the line through (2,4) and parallel to y=4x+2


http://www.webmath.com/equline2.html




Enter as (2,4) and slope is 4

ANSWER : f(x)=4x-4

Find an equation of the line.

Slope = -2; through (-3,5)


Webmath shows answer as y=2x-1

Help shows answer as 2X+y=-1


Slope = 4; through (-5,8)


Webmath shows answer as y=4x+28

Help shows answer as 4x-y=-28


Slope = 3; through (-5,8)


Webmath shows answer as y=3x+23

Help shows answer as 3x-y=-23

Write the equation of the line in standard form

With slope -5/7; y-intercept 5

First…WEBMATH: http://www.webmath.com/equline3.html

Then take that solution and go to: http://www.algebra.com/algebra/homework/Linear-equations/converting-linear-equations.solver and convert from slope intercept to standard.

5x+7y=35

http://www.algebra.com/algebra/homework/Linear-equations/converting-linear-equations.solver






Find the equation of the line through (-6,-2) and (0,-1). Write the equation in standard form.

First…WEBMATH: http://www.webmath.com/equline3.html

Then take that solution and go to: http://www.algebra.com/algebra/homework/Linear-equations/converting-linear-equations.solver and convert from slope intercept to standard.

Webmath says: y=1/6x-1 (since algebra.com doesn’t have “subtract” I entered it as y=/1/6x+-1 to which

ALGEBRA.com says: 1x+6y=-6

Correct answer: x-6y=6 so pick answer most similar to what algebra turns in for minus questions

Find an equation of the line graphed. See graph and write the ordered pair with x,y intercepts. Example: (4,3) and (-2,0)

Solve at WEBMATH: http://www.webmath.com/equline1.html

Solve. Write the solution in interval notation.

MATHWAY – Convert to set notation

ANSWER:

Solve. (MATHWAY – Convert to set notation)

2x-17> -5

ANSWER:

Solve. (MATHWAY – Convert to set notation)

3x-5> 1

ANSWER:





The graph to the right shows the research and development expenditures by the Acme Medicine Company as a function of time. The function below approximates the graph.

F(x)=1.558x-3092

1. Use the graph to approximate the money spent on research and development in 1999.

SOLUTION: Simply look at graph and see where it corresponds. $23 billion

2. Use the function to approximate the money spent on research and development in 1999.

SOLUTION: replace the x with the year in question. F(1999)=1.558(1999)-3092 f(1999)= $22.442 billion

Use the graph of the function shown to the right to find all values of x such that g(x)=8.

Look at graph and on the y-line, find where the noodle line crosses and the answer would look like

X=-5, -1, 2

Use the graph of the function to find all values of x such that f(x)=3.

Look at graph and see where in this case 3 is then look across to where the upside down U is. Draw an imaginary line across and the answer looks like

X=-1,3

Use the graph of the function f to find f(-3). Line looks like below. Find -3 on x line then look down to where it crosses the y line. That’s the answer. Looks like f(-3)=-3

The function f(x)=0.36x+12.6 can be sued to predict diamond production. For this function, x is the number of years after 2000, and f(x) is the value (in billions of dollars) of the year’s diamond production. Use this function to predict diamond production in 2006.

f(6)=0.36(6)+12.6

ANSWER: f(6) = 14.76 billion

The function A(r) = pi(r)^2 may be used to find the area of a circle with radius r. Find the area of a circle whose radius is 2 centimeters.

A(2) = pi (2)^2

Basically to solve take 2^2 = 4pi (make sure to include pi symbol)

The function V(x) = x^3 may be used to find the volume of a cube with side length x. Find the volume of a cube whose side is 10 centimeters.

The volume is ___ cubic centimeters.

To solve V(10)=(10)^3

10^3=1000

ANSWER: The volume is 1000 cubic centimeters.

Forensic scientists use the following function to find the height of a woman if they are given the height of her femur bone, f, in centimeters. Find the height of the woman whose femur measures 59 centimeters.

H(f)=2.59f+47.24

To calculate in Mathway, simplify 2.59(59)+47.24 = 200.05 centimeters

The dosage in milligrams D of a heartworm preventive for a dog weighs x pounds is given by D(x)=136/25x. Find the proper dosage for a dog that weighs 27 lbs.

Simplify 136/25*(27)=146.88

The dosage is 146.88 miligrams

The dosage in milligrams D of a heartworm preventive for a dog weighs x pounds is given by D(x)=136/25x. Find the proper dosage for a dog that weighs 26 lbs.

Simplify 136/25*(26)=141.44 milligrams

The dosage is 141.44 miligrams

Solve the system of equations by graphing.

-x+y=2

Y=3

Enter in mathway as is above and “solve by the substitution method”

Without actually solving the problem, choose the correct solution by deciding which choice satisfies the given conditions. Kesha has a total of 120 coins, all of which are dimes or quarters. The total value of the coins is $16.50. Find the number of each type of coin.

Which choice satisfies the given condition?

a. 90 dimes, 30 quartersb. 80 dimes, 40 quartersc. 20 dimes, 58 quarters

ANSWER : Add them up and see what the total is. In this case it’s A.

Write a system of equations in x and y describing the situation. Do not solve the system.

Keiko has a total of $5800, which she invested in two accounts. The larger account is $900 greater than the smaller account. Let X be the amount of money in the larger account and y be the amount of money in the smaller account.

The total ( sum ) is 5800

X+y = 5800

The larger account is $900 greater than the smaller

X = y+900

Thus the system of equations is:

X+y=5800X=y+900

Norman and Suzanne own 29 shares of a fast food restaurant stock and 75 shares of a toy company stock. At the close of the markets on a particular day in 2004, their stock portfolio consisting of these two stocks was worth 1297.00. The closing price of the fast food stock was $25 more per share than the closing price of the toy company stock on that day. What was the closing price of each stock on that day?

The value of 35 shares and the value of 73 was $1297.00 Of the fast food shares of the toy worthRestaurant stock company stock 29X + 75y = $1297.00

The closing price was $25 more per share than the closing Of the fast food price of the toy company stock on thatRestaurant stock day. X = 25 + Y

29x+75y=1297X=25+Y

NOW….go to http://www.calculatorsoup.com/calculators/conversions/fractiontodecimal.php

to simplify the fractions and your answer is:

y= 5.50 toy stockx=30.50 fast food stock

Doreen Schmidt is a chemist. She needs to prepare 24 ounces of a 10% hydrochloric acid solution. Find the amount of 12% solution and the amount of 6% solution she should mix to get this solution.

Concentration rate Ounces of solution Ounces of pure acidFirst Solution 6% X 0.6xSecond Solution 12% Y 0.12yMixture Needed 10% 24 0.10(24)

Ounces of 9% and Ounces of 18% = 24

x + y = 24

acid in 6% and acid in 12% = acid in mixture

0.06x + 0.12y = 2.4 (which is mixture needed 0.15*24=2.4)

So find the solution:

X+y=24

http://www.calculatorsoup.com/calculators/conversions/fractiontodecimal.php

0.06x+0.12y=2.4

X=8 (ounces of 6%)

y=16 (ounces of 12%)

Example: Doreen is a chemist. She needs to prepare 36 ounces of a 15% hydrochloric acid solution. Find the amount of 18% solution and the amount of 9% solution she should mix to get this solution.



x + y = 36




X+y=36

0.09x+0.18y=5.4

X=12, y=24

Example: Doreen is a chemist. She needs to prepare 36 ounces of a 17% hydrochloric acid solution. Find the amount of 18% solution and the amount of 9% solution she should mix to get this solution.



x + y = 36




X+y=36

0.09x+0.18y=6.12

X=4, y=32

Gabe Amodeo, a nuclear physicist, needs 60 liters of a 70% acid solution. He currently has a 50% solution and a 80% solution. How many liters of each does he need to make the needed 60 liters of 70% acid solution.

Concentration rate Liters of Solution Liters of Pure Acid

First Solution 50% x 0.50x

Second Solution 80% y 0.80y

Mixed Solution 70% 60 (60)(0.70) (answer: 42)

Liters of 50% solution + Liters of 80% solution = 60

X + Y = 60

Acid in 50% solution + acid in 80% solution = (60)(0.70)

0.50x + 0.80y = (60)(0.70)

Written as:

X+y=600.50x+0.80y=42

X=20Y=40(20,40)

ANSWER: Gabe will need 20 liters of the 50% solution and 40 liters of the 80% solution.

Gabe Amodeo, a nuclear physicist, needs 60 liters of a 40% acid solution. He currently has a 20% solution and a 50% solution. How many liters of each does he need to make the needed 60 liters of 40% acid solution.

Concentration rate Liters of Solution Liters of Pure Acid

First Solution 20% x 0.20x

Second Solution 50% y 0.50y

Mixed Solution 40% 60 (60)(0.40) (answer: 24)

Liters of 40% solution + Liters of 40% solution = 60

X + Y = 60

Acid in 20% solution + acid in 50% solution = (60)(0.40)

0.20x + 0.50y = (60)(0.40)

Written as:

X+y=600.20x+0.50y=24

X=20Y=40(20,40)

ANSWER: Gabe will need 20 liters of the 20% solution and 40 liters of the 50% solution.

Matrice and matrix:

x + 2y + 3z = 9

2x - y + z = 8

3x - z = 3

This would be entered into the matrix above as :

1 2 3 : 9

2 -1 1 : 8

3 0 -1 : 3

For the following system, what is the correct corresponding matrix?

X + Z = 5Y + 2z = -3

2x - Y = 13

ANSWER:1 0 1 50 1 2 -32 -1 0 13

Solve linear equality:

Choose the correct interval notation for the solution set.

ANSWER: In mathway, enter as above then choose “Convert to Set Notation”

Dale and Sharon Mahnke have decided to fence off the garden plot behind their house, using their house as a “fence” along one side of the garden. The length (which runs parallel to the house) is 4 feet less than twice the width. Find the dimensions if 32 feet of fencing is used along the three sides requiring it.

The length is 3 feet less than twice the width L = 2W-432 feet of fencing is used along three sides 32 = 2W+L

L=2W-432=2W+L

L=14 (length)W=9 (width)

Dale and Sharon Mahnke have decided to fence off the garden plot behind their house, using their house as a “fence” along one side of the garden. The length (which runs parallel to the house) is 3 feet less than twice the width. Find the dimensions if 25 feet of fencing is used along the three sides requiring it.

The length is 3 feet less than twice the width L = 2W-325 feet of fencing is used along three sides 25 = 2W+L

L=2W-325=2W+L

L=11 (length)W=7 (width)

Use the graph of the function f to find f(2). Graph has noodle like .To find f(2) locate 2 on the x-axis.The x-vale of 2 will correspond to where the noodle crosses the line at 2. ANSWER f(2)=-1

Use the graph of the function to find all values of x such that f(x)=-3Draw a straight line left to right where -3 is and note where the line crosses. Answer: x=-2,4

If A = {x|x is an even integer}, B={x|x is an odd integer}, C={2,3,4,5} and D={5,6,7,8}, list the elements of the following sets.

CUD

Answer: CUD={2,3,4,5,6,7,8}

If A = {x|x is an even integer}, B={x|x is an odd integer}, C={2,3,4,5} and D={5,6,7,8}, list the elements of the following sets.

CUD

Answer: CUD={2,3,4,5,6,7,8}

A D

Since A is even integers we’d pick the even one’s out of D={6,8}

Solve the compound inequality. Graph the solution set and write it in interval notation.

X<0 and x>4

In mathway “find the intersection of the equalities”

Written in interval notation (-2,4) by the way this is NOT the correct answer to above.

Solve the compound inequality. Graph the solution set and write in interval notation.

X<-4 or x>2 (In mathway, “find the union of the inequalities)

U

Make sure to include the U in the answer

Solve the compound inequality. Graph the solution set and write in interval notation.x>1 or x<7

U Enter answer as (-sideways 8, sideways 8)

Write an absolute value equation representing all numbers x whose distance from 0 is 19 units.

|x|=19

Write an absolute value equation representing all numbers x whose distance from 0 is 25 units.

|x|=25

Solve the inequality. Then graph the solution set.

|x|<1

In mathway, choose “convert to set notation”

[-1,1]

Solve the inequality. Then graph the solution set and write it in interval notation.

|-5x|>-6

In mathway, choose “convert to set notation”

(-sideways8,sideways 8) don’t ask me why because mathway says no solution.

When graph the inequality x+y<-5 Remember to SHADE the half after you graph it.

Find the root is 1.

Simplify

(Hint: Factor the polynomial first)

Answer x+2 – since this is PROBABLY a perfect square answer as |x+2|

If f(x)=square root 2x+3, find f(-1)

Replace x in the function with -1

F(-1)=square root 2(-1)+3

In mathway “evaluate the function”

If f(x)= , find f(-1)

When calculating in mathway, remove the f(x) and just calculate the straight square root

Which of the following is closest to?

10 12 20 30

Simplify in mathway and answer is:

Exact Solution

Approximate Solution: 12.0416

Answer: 12

9 Evaluate

Remember that a is the same as =

So 9 =

= 3

36 Evaluate


So 36 = = 6

Use radical notation to write the expression. Assume that all variables represent positive real numbers.

2m = 2*(m )

Type as a radical based on the definition of r .

2*(m ) =

Use radical notation to write the expression. Assume that all variables represent nonnegative quantities.

(121x^4) Evaluate


So (121x^4) =

=

(16x^4) Evaluate


So (16x^4) =

=

Use radical notation to write the expression.

(-343) =

Write with positive exponents. Simplify.

= b^2/5

A zorb is a large inflated ball within a ball. The formula for the radius of a sphere with surface area A is given

by r= . Calculate the radius of a standard zorb whose outside surfce area is 60.82 sq m.

Calculate as

Exact solution 2.2

A zorb is a large inflated ball within a ball. The formula for the radius of a sphere with surface area A is given

by r= . Calculate the radius of a standard zorb whose outside surfce area is 15.21 sq m.

Calculate as

Exact solution 1.1

Find the perimeter of the trapezoid. Add all the ones and simplify.

A 34 inch TV is advertised in the local newspaper. If 34 inches is the measure of the diagonal of the picture tube, find the measure of each side of the picture tube.

Don’t know this one

The graph to the right shows the daily low temperatures for one week in a city. Notice that the shape of the temperature graph is similar to the curve drawn. In fact, this graph can be modeled by the quadratic function

F(x)=3x^2-18x+77, where f(x) is the temperature in degrees Farenheit and x is the number of days from Sunday. Use the quadratic function to approximate the temp on Wednesday.

Answer: Count how many days from Sunday is the day in question. Since Wednessday is 3 days from Sunday you’d replace x with 3 in the equation. “Evaluate the function”

The Soule’s live on a corner lot. Often, children cut across their lot to save walking distance.

X+35 85 ft

x

Use the pythagorean theorum that says “In a right angled triangle: the square of the hypotenuse is equal to the sum of the squares of the other two sides.”

X^2+(x+35)^2=85^2

X= -75, 40

Since it can’t be negative it has to be 40. If x = 40 then the x+35 side must be 75 feet. 40 + 75 feet equals 115 feet if the children to walk AROUND the block. 115-85=30 feet of walking distance saved.

The Soule’s live on a corner lot. Often, children cut across their lot to save walking distance.

X+7 17 ft

x

Use the pythagorean theorum that says “In a right angled triangle: the square of the hypotenuse is equal to the sum of the squares of the other two sides.”

X^2+(x+7)^2=17^2

X= -15,8

Since it can’t be negative it has to be 8. If x = 8 then the x+7 side must be 15 feet. 8 + 15 feet equals 23 feet if the children to walk AROUND the block. 23-17=6 feet of walking distance saved.

Write as an exponential equation. In Mathway, enter as:

ANSWER Make sure to write it that way

INFO From http://webgraphing.com/graphing_advanced.jsp#

Entering Intermediate and Advanced Functions

Mathematical Form Input (Keyboard) Form|x| abs(x)

sqrt(x)

logex log(x) or ln(x) or log(e,x)log5x log(5,x)tan x tan(x)

sin 2πx sin(2pix) or sin(2pi*x)

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Solve the equation for x. Give an exact solution and a four-decimal-place approximation.Log x=3.1

Remember, the base is understood to be 10 so rewrite as follows:10^3.1=xAnd calculate to “solve the equation for x”10^3.1=1258.9254

Solve the equation for x. Give an exact solution. ENTER INTO WOLFRAM:Ln x=-1.4The exact answer is x=e^-1.4Then calculate that in WOLFRAM: 0.2466

Solve the equation for x. Give an exact solution. ENTER INTO WOLFRAM:Ln 5x=4The exact answer is x=e^4/5Then calculate that in WOLFRAM: 10.9196

Write the first five terms of each sequence whose general term is given.

a[n]=2n+5

a1= what? Simply replace the n’s with 1 and solve the equation for A

a2= what? Simply replace the n’s with 2 and solve the equation for A. etc etc

Evaluate the series. Sigma Series.

3

(-2i+11)

i=1

Here’s how to compute it:

[-2(1)+11]+[-2(2)+11]+[-2(3)+11]

ANSWER=21

Here's the official notation and don't worry -- I'll explain each part:

E

* Notice that we're using a k instead of the n... This is important and will make something easier later.

Evaluate the series. Sigma Series.

6

(2i+1)

i=1

Here’s how to compute it:

[2(1)+1]+[2(2)+1]+[2(3)+1]+[2(4)+1]+[2(5)+1]+[2(6)+1]

ANSWER=48

E

Jen has been pricing speed pass train fares for a group trip. Three adults and four children must pay $136. Two adults and three children must pay $97. Find the price of the adult and the child ticket.

A=the price of an adults ticket

B=the price of a child’s ticket

Three adults and four children must pay $136

3A + 4C = $136

Two adults and three children must pay $97

2A + 3C = $97

Add the two and solve using Addition/Elimination Method as below:

Solve by Addition/Elimination Method:

3a+4c=136 2a+3c=97

Answer:

c=19 a=20

To mix weed killer with water correctly, it is necessary to mix 8 teaspoons of weed killer with 2 gallons of water.Find how many gallons of water are needed to mix with the entire box if it contains 36 teaspoons of weed killer.:Let x = amt of water for 36 tsp of weed killer:Use a ratio equation

= 8x = 2*368x = 72

x = x = 9 gallons of water

EXAMPLE:

To mix weed killer with water correctly, it is necessary to mix 20 teaspoons of weed killer with 5 gallons of water.Find how many gallons of water are needed to mix with the entire box if it contains 32 teaspoons of weed killer.:Let x = amt of water for 32 tsp of weed killer:Use a ratio equation

= x/5 = 32/20x = 8 gallons of water

dmat 0093 notes

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