Examkin

Question

Difficulty: MediumLinear Functions and Graphs

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)
335454
557878
88114114

Which of the following is the total cost, in dollars, of renting the concrete mixer for 1212 hours?

  1. A
    5555
  2. B
    144144
  3. 162162Answer
  4. D
    168168

Answer

162162 dollars
The cost function is linear, represented by C(t)=mt+bC(t) = mt + b. The rate of change (slope) is calculated as m=78−545−3=12m = \frac{78 - 54}{5 - 3} = 12 dollars per hour. Using the point (3,54)(3, 54), we solve for the initial fee (y-intercept) bb: 54=12(3)+b54 = 12(3) + b, which gives b=18b = 18. Thus, the linear model is C(t)=12t+18C(t) = 12t + 18. For a rental time of 1212 hours, the total cost is C(12)=12(12)+18=162C(12) = 12(12) + 18 = 162 dollars.

Step-by-Step Solution

1
Calculate the rate of change (slope) of the linear cost function.
m=12m = 12
Since the relationship is linear, the rate of change is constant. Using the coordinate points (3,54)(3, 54) and (5,78)(5, 78) from the table, the slope is calculated as m=78−545−3=12m = \frac{78 - 54}{5 - 3} = 12.
2
Determine the y-intercept (initial fee) of the function.
b=18b = 18
Substitute one of the points, such as (3,54)(3, 54), and the slope m=12m = 12 into the slope-intercept form equation C(t)=mt+bC(t) = mt + b. This gives 54=12(3)+b⇒54=36+b⇒b=1854 = 12(3) + b \Rightarrow 54 = 36 + b \Rightarrow b = 18.
3
Evaluate the cost function for a rental time of 1212 hours.
C(12)=162C(12) = 162
Using the complete linear equation C(t)=12t+18C(t) = 12t + 18, substitute t=12t = 12 to find the total cost: C(12)=12(12)+18=144+18=162C(12) = 12(12) + 18 = 144 + 18 = 162.

Key Concept

Finding and evaluating a linear function from tabular data
Estimated Time:1m 30s
Rate this question

Topics

Linear Functions and Graphs

More questions from this topic