EVALUATING REAL WORLD EXPRESSIONS

Example 1 : 

A scientist measures the air temperature in Death Valley, California, and records 50 °C. The expression

1.8c + 32

gives the temperature in degrees Fahrenheit for a given temperature in degrees Celsius c. Find the temperature in degrees Fahrenheit that is equivalent to 50 °C.

Solution : 

Step 1 :

Fromm the given information, we have 

c  =  50 °C

Step 2 :

Plug the value into the expression.

1.8c + 32

Plug 50 for c 

1.8(50) + 32

Multiply 

90 + 32

Add

122

Hence, 122 °F is equivalent to 50 °C.

Example 2 : 

The expression 6x² gives the surface area of a cube, where x is the length of one side of the cube. Find the surface area of a cube with a side length of 2 meters.

Solution : 

Step 1 :

Fromm the given information, we have 

x  =  2

Step 2 :

Plug the value into the expression which represents surface area of the cube.

6x²

Plug 2 for x 

6(2)²

Evaluate exponents

6(4)

Multiply 

24

Hence, the surface area of the cube 24 m²

Example 3 : 

The expression x³ gives the volume of a cube, where x is the length of one side of the cube. Find the volume of a cube with a side length of 2 meters.

Solution : 

Step 1 :

Fromm the given information, we have 

x  =  2

Step 2 :

Plug the value into the expression.

x³

Plug 2 for x 

(2)³

Evaluate exponents 

8

Hence, the volume of the cube 8 m³

Example 4 : 

The expression 60m gives the number of seconds in m minutes. How many seconds are there in 7 minutes?

Solution : 

Step 1 :

Fromm the given information, we have 

m  =  7

Step 2 :

Plug the value into the expression.

60m

Plug 7 for m 

60(7)

Multiply 

420

Hence, there are 420 seconds in 7 minutes. 

Example 5 : 

A right circular cylinder has a volume of 377 cubic centimeters. The area of the base of the cylinder is 13 square centimeters. What is the height, in centimeters, of the cylinder?

Solution :

Volume of 3D shape = base area x height

Volume of right circular cylinder = 377 cubic centimeters

base area = 13 square cm

height = h

377 = 13 x h

h = 377/13

h = 29

So, the required height of the cylinder is 29 cm.

Example 6 : 

A circle has a radius of 43 meters. What is the area, in square meters, of the circle?

Solution :

Area of circle = πr²

radius = 43 m

= π(43)²

= 1849π cm²

Example 7 : 

The expression 20a + 13c is the cost for a adults and  c students to enter the Miami Science Museum.

a. Find the total cost for 4 adults and 24 students.

b. You find the cost for a group. Then the numbers of adults and students in the group both double. Does the cost double?

Explain your answer using an example. c. In part (a), the number of adults doubles, but the number of students is cut in half. Is the cost the same? Explain your answer using an example.

Solution :

20a + 13c ----(1)

Number of adults = 4, number of students = 24

applying these values in the expression above, we get

= 20(4) + 13(4)

= 80 + 52

= 132

b) Number of adults double, then a = 2a

Number of students double, then c = 2c

= 20(2a) + 13(2c)

= 20a + 26c

=2(10a + 13c)

So, the total cost doubled

c) Number of adults = 2a

Number of students = c/2

Applying these values in (1), we get

= 20(2a) + 13(c/2)

= 40a + 13c/2

= (80a + 13c)/2

Cost will not be the same.

Example 8 : 

After m months, the length of a fingernail is 10 + 3m millimeters. How long is the fingernail after eight months? three years?

Solution :

Length of fingernail = 10 + 3m

Length of fingernail after 8 months 

m = 8

= 10 + 3(8)

= 10 + 24

= 34 mm

Number of months in a year = 12

Number of months in 3 years = 3(12)

= 36 years

= length of fingernail = 10 + 3(36)

= 10 + 108

= 118 mm

So, the length of fingernail after 8 months is 34 mm

Length of fingernail after 3 years is 118 mm.

Example 9 :

You earn 15n dollars for mowing n lawns. How much do you earn for mowing one lawn? seven lawns?

Solution :

Amount earning = 15n

Amount he is earning for moving one lawn :

= 15(1)

= $15

Amount he is earning for moving seven lawns :

= 15(7)

= $105

Example 10 :

You are saving for a skateboard. Your aunt gives you $45 to start and you save $3 each week. The expression

45 + 3w

gives the amount of money you save after w weeks.

a. How much will you have after 4 weeks, 10 weeks, and 20 weeks?

b. After 20 weeks, can you buy the skateboard? Explain.

Solution :

a)

b)  After 20 weeks, you have $105. So, you cannot buy the $125 skateboard

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