chapter 2 equations and inequalities in one variable section 7 problem solving: geometry and uniform...

Chapter 2

Equations and Inequalities in One Variable

Section 7

Problem Solving: Geometry and Uniform

Motion

Sullivan & Struve, Elementary and Intermediate Algebra, 2/e 2.7 - 2© 2010 Pearson Education, Inc.

Section 2.7 Objectives

1 Set Up and Solve Complementary and

Supplementary Angle Problems

2 Set Up and Solve Angles of Triangle Problems

3 Use Geometry Formulas to Solve Problems

4 Set Up and Solve Uniform Motion Problems


Angles

Two angles whose sum is 90° are called complementary angles. Each angle is called the complement of the other.

Two angles whose sum is 180° are called supplementary angles. Each angle is called the supplement of the other.

(90 – x)°x°

complementary angles

x°(180 – x)°

supplementary angles


Solving Angle Problems

Example:

Angle A and angle B are complementary angles, and angle A is 21º more than twice angle B. Find the measure of both angles.

Step 1: Identify This is complementary problem. We are looking for the measure of the two angles whose sum is 90°.

Step 2: Name Let a represent the measure of angle A.

B

A

Continued.



Example continued:

Step 3: Translate Angle A is 21° more than twice the measure of angle B.

a = 21 + 2 · m B

a = 21 + 2 (90 – a)

Continued.

Step 4: Solve Solve the equation.

a = 21 + 2 (90 – a)

a = 21 + 180 – 2a Distribute.

a = 201 – 2a Combine like terms.



Example continued:

a = 201 – 2a

3a = 201 Add 2a to both sides.

a = 67 Divide both sides by 3.

Step 5: Check The measure of A is 67°. The measure of B is 90° – a = 90° – 67° = 23°.

Step 6: Answer the Question

67° + 23° = 90°

The two complementary angles measure 67° and 23°.


Solving Triangle Problems

Example:

Find the measure of Angle C.

Step 1: Identify This is an “angles of the triangle” problem.

Step 2: Name Let c represent the measure of angle C.

Continued.

42°84°A B

C

Remember that the sum of the measures of the interior angles of a triangle is 180°.



Example continued:

Continued.

Step 3: Translate The three angles add up to 180°.


84 + 42 + c = 180

84 + 42 + c = 180

126 + c = 180 Combine like terms.

c = 54 Subtract 126 from both sides.



Example continued:

84° + 42° + 54° = 180°

Step 5: Check Is the sum of the three angles equal to 180°?

180° = 180°

Step 6: Answer the Question The measure of Angle C is 54°.

42°84°A B

C

54°


Solving Geometry Problems

Example:Julie is making cone-shaped candles. The mold for the candles is 4 in. in diameter and 7 in. high. How many cubic inches of wax does Julie need to buy if she wants to make 50 candles?

Step 1: Identify This is a geometry volume problem. We want to find the volume of the cone-shaped candle to determine the amount of wax needed for 50 candles.

Step 2: Name Let V represent the volume of one cone.

Continued.

2

3r hV


Solving Geometry Problems

Example continued:

2

3r hV

Continued.

Step 3: Translate We need to use the formula for the volume of a cone.


2(3.14)(2 in.) (7 in.)3

3.14; r = 2; h = 7

2(3.14)(2 in.) (7 in.)3

V 3(3.14)(28 in. )

3

387.92 in.3


Solving Geometry ProblemsExample continued:

387.92 in.3

V 329.3 in. This is the amount needed for one candle.

29.3 in.3 50 1465 in.3 This is the amount needed for 50 candles.

Step 5: Check

Step 6: Answer the Question Julie needs to buy approximately 1465 in.3 of wax to make 50 candles.

23(3.14)(2 in.) (7 in.) 29.3 in.

3V

29.3 in.3 50 = 1465 in.3


Uniform MotionObjects that move at a constant velocity (speed) are said to be in uniform motion.

Uniform Motion Formula

If an object moves at an average speed r, the distance d covered in time t is given by the formula

d = rt.

Rate · Time = Distance

Object #1 distance 1

Object #2 distance 2

The following table is helpful in solving motion problems.


Uniform Motion Problem

Example:Steve jogs at an average rate of 8 kilometers per hour. How long would it take him to jog 14 kilometers?

Step 1: Identify This is a uniform motion problem. We are looking for the length of time it would take Steve to jog 14 kilometers.

Step 2: Name Let t represent the length of time it would take Steve to jog 14 kilometers.

Continued.



Example continued:

Step 4: Solve

Continued.

Divide both sides by 8.

Step 3: Translate Organize the information in a table.

Distance, km Rate, kph Time, hours

14 8 t

14 = 8td = rt

148

t

1.75t

14 8t

Simplify.



Example continued:

Step 5: Check t = 1.75 represents the length of time.


14 = (8)(1.75)

It takes Mark 1.75 hours (or 1 hour and 45 minutes) to run 14 kilometers.

d = rt

14 = 14



Example:Nina drove her car to Cleveland while Paula drove her car to Columbus. Nina drove 360 kilometers while Paula drove 280 kilometers. Nina drove 20 kilometers per hour faster than Paula on her trip. What was the average speed in kilometers per hour for each driver?

Let r = the rate of Paula’s car.

Step 1: Identify

Continued.

Let r + 20 = the rate of Nina’s car.

Distance problems can be solved using the formuladistance = rate × time (d = rt).

dtr

The time, t, for each driver was the same.

Step 2: Name



Continued.

Step 4: Solve

Example continued:

r280Paula

r + 20360Nina

rd dtr

36020r

280r

360 28020r r

Since the time for each driver was the same, we can set the times equal to each other.

Step 3: Translate


Uniform Motion ProblemExample continued:

Multiply both sides by r(r + 20).

360 28020r r

360 280

20( 20) ( 20)r r r

r rr

Simplify.360 280( 20)r r

Subtract 280r from each side.

360 280 5600r r

Paula’s rate was 70 kilometers per hour.

Nina’s rate was r + 20 = 90 kilometers per hour.

Divide each side by 80.70r

Distribute.

80 5600r

Continued.


Uniform Motion ProblemExample continued:

Step 5: Check360 280

20r r

360 2802070 70

360 28090 70

4 4


Paula’s rate was 70 kilometers per hour.

Nina’s rate was 90 kilometers per hour.

chapter 2 equations and inequalities in one variable section 7 problem solving: geometry and uniform...

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