Question 637·Medium·Algebra

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A delivery service charges a fixed fee plus a constant fee per mile driven. A customer is charged $13 for a 6-mile trip and $25 for a 14-mile trip.

Assuming the cost $C$ , in dollars, is a linear function of the distance $d$ , in miles, what is the fixed fee the service charges?

Your answer

A fixed fee plus a per-mile charge signals a linear model: total cost equals the mileage charge plus a fee paid once. Use the two trip totals to find the cost per mile, then subtract the mileage charge from one trip’s total. A trip’s total is not the fixed fee unless its distance is zero.

Hints

Hint 1
In a linear model, write total cost as $C(d)=md+b$ . The term $md$ changes with the miles driven, while $b$ is paid once. Which term is the fixed fee?
Hint 2
The slope is the added cost per mile. Divide the change in cost between the two trips by the change in miles, keeping dollars on top and miles on the bottom.
Hint 3
A trip’s total includes both the mileage charge and the fixed fee. Once you know the cost per mile, subtract the mileage charge from either trip’s total.

Step-by-step

Find the per-mile cost, then the fixed fee
Step 1
Model the two parts of the cost
Write $C(d)=md+b$ , where $m$ is the dollars charged per mile and $b$ is paid once. At $d=0$ , the mileage charge $md$ is $0$ , so $b$ is the fixed fee, also called the intercept.
Step 2
Find the cost per mile
The slope $m$ is the change in cost divided by the change in miles. Type $\frac{25-13}{14-6}$ into Desmos. It shows $1.5$ , so the per-mile charge is $1.5$ dollars, not the fixed fee.
Step 3
Remove the mileage charge
The 6-mile total is $13=1.5(6)+b$ , so subtract its mileage charge to find $b$ . Type $13-1.5(6)$ into Desmos; it shows $4$ . The fixed fee is $4. Grid in 4.

Question 637·Medium·Algebra

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