A construction company rents a concrete mixer. The total cost, in dollars, of renting the … — SAT

A construction company rents a concrete mixer. The total cost, in dollars, of renting the concrete mixer is a linear function of the rental time, in hours. The table below shows some values of the rental time and the corresponding total cost.

Time (hours)	Total cost (dollars)
$3$	$54$
$5$	$78$
$8$	$114$

Which of the following is the total cost, in dollars, of renting the concrete mixer for $12$ hours?

A
$55$
B
$144$
$162$ Cevap
D
$168$

Cevap

162

dollars

The cost function is linear, represented by

C(t) = mt + b

. The rate of change (slope) is calculated as

m = \frac{78 - 54}{5 - 3} = 12

dollars per hour. Using the point

(3, 54)

, we solve for the initial fee (y-intercept)

b

54 = 12(3) + b

, which gives

b = 18

. Thus, the linear model is

C(t) = 12t + 18

. For a rental time of

12

hours, the total cost is

C(12) = 12(12) + 18 = 162

dollars.

Adım Adım Çözüm

Calculate the rate of change (slope) of the linear cost function.

m = 12

Since the relationship is linear, the rate of change is constant. Using the coordinate points

(3, 54)

and

(5, 78)

from the table, the slope is calculated as

m = \frac{78 - 54}{5 - 3} = 12

Determine the y-intercept (initial fee) of the function.

b = 18

Substitute one of the points, such as

(3, 54)

, and the slope

m = 12

into the slope-intercept form equation

C(t) = mt + b

. This gives

54 = 12(3) + b \Rightarrow 54 = 36 + b \Rightarrow b = 18

Evaluate the cost function for a rental time of

12

hours.

C(12) = 162

Using the complete linear equation

C(t) = 12t + 18

, substitute

t = 12

to find the total cost:

C(12) = 12(12) + 18 = 144 + 18 = 162

Anahtar Kavram

Finding and evaluating a linear function from tabular data

Tahmini Süre:1m 30s

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