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Difficulty: MediumLinear Functions and Graphs

Company A and Company B both rent moving trucks. The total cost A(d)A(d), in dollars, for renting a truck from Company A for dd days is given by A(d)=35d+75A(d) = 35d + 75. The table below shows some values of the total cost B(d)B(d), in dollars, for renting a truck from Company B for dd days.

Days (dd)Total Cost (B(d)B(d))
2150
5300

If the relationship between dd and B(d)B(d) is linear, for how many days of rental will the total cost at Company B be exactly 9595 dollars more than the total cost at Company A?

  1. A
    2
  2. B
    3
  3. C
    5
  4. 8Answer

Answer

The total cost at Company B will be exactly 95 dollars more than the total cost at Company A for a rental of 8 days.
To find the number of days for which Company B's cost is exactly 95 dollars more than Company A's cost, we first find the linear function for Company B. The rate of change is 300−1505−2=50\frac{300 - 150}{5 - 2} = 50 dollars per day. The initial cost is 150−50(2)=50150 - 50(2) = 50 dollars, giving B(d)=50d+50B(d) = 50d + 50. We then set up the equation B(d)−A(d)=95B(d) - A(d) = 95, which is (50d+50)−(35d+75)=95(50d + 50) - (35d + 75) = 95. Simplifying this equation gives 15d−25=9515d - 25 = 95. Adding 25 to both sides gives 15d=12015d = 120, and dividing by 15 gives d=8d = 8.

Step-by-Step Solution

1
Determine the linear cost function for Company B, B(d)=md+bB(d) = md + b, using the points (2,150)(2, 150) and (5,300)(5, 300) from the table.
The slope is m=300−1505−2=50m = \frac{300 - 150}{5 - 2} = 50. Using (2,150)(2, 150) to find the y-intercept: 150=50(2)+b⇒b=50150 = 50(2) + b \Rightarrow b = 50. Thus, B(d)=50d+50B(d) = 50d + 50.
We need to find the equation representing the total cost of Company B to set up the comparison.
2
Set up an equation where the total cost at Company B is equal to the total cost at Company A plus 95 dollars.
B(d)=A(d)+95⇒50d+50=(35d+75)+95B(d) = A(d) + 95 \Rightarrow 50d + 50 = (35d + 75) + 95.
This represents the condition that Company B's cost is 95 dollars more than Company A's cost.
3
Simplify the equation and solve for the number of days, dd.
50d+50=35d+170⇒15d=120⇒d=850d + 50 = 35d + 170 \Rightarrow 15d = 120 \Rightarrow d = 8.
Isolating the variable dd gives the exact number of days required.

Key Concept

Comparing linear functions and solving linear equations representing real-world contexts.
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Topics

Linear Functions and Graphs

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