Question 14·Medium·Linear Functions
An app-based car service charges a fixed booking fee plus a constant rate per mile. A 3-mile trip costs $11, and an 8-mile trip costs $26.
What is the total cost of a 5-mile trip?
For “fixed fee plus constant rate” problems, think of a linear cost function. First, ignore the fixed fee and use the two given trips to find the rate per mile by dividing the change in cost by the change in miles. Then, instead of solving for the fee explicitly, move from a known trip to the requested distance by adding (or subtracting) the rate times the mile difference. This avoids extra algebra and gives the answer quickly while still respecting the constant-rate condition.
Hints
Identify the pattern
The service has a fixed fee plus a constant rate per mile. How can you use the difference between the two given trips to find the rate per mile, without worrying about the fixed fee at first?
Use the two trips to find the rate
Compare the change in cost to the change in miles from the 3-mile trip to the 8-mile trip. What do you get if you divide the cost difference by the mile difference?
Move from 3 miles to 5 miles
Once you know how many dollars each extra mile costs, how much more should a 5-mile trip cost than a 3-mile trip? Add that amount to the 3-mile cost.
Desmos Guide
Compute the 5-mile cost with one expression
In a new expression line in Desmos, type:
11 + (26 - 11)/(8 - 3) * (5 - 3)
Desmos will output a single number; that value is the cost of a 5-mile trip according to the fixed-fee, constant-rate pattern.
Step-by-step Explanation
Find the rate per mile
Because the service charges a fixed booking fee plus a constant rate per mile, the difference in cost between two trips comes only from the miles driven.
From 3 miles to 8 miles:
- Cost increases from $11 to $26, a change of $26 - 11 = 15.
- Distance increases from 3 miles to 8 miles, a change of miles.
So the rate per mile is
Express the cost of a 5-mile trip using the rate
Use the known 3-mile trip and the rate you just found.
From 3 miles to 5 miles:
- The distance increases by miles.
- Each extra mile costs dollars.
So the extra cost from 3 miles to 5 miles is
The cost for 5 miles is therefore
Compute and match to the answer choices
Now evaluate the expression:
So a 5-mile trip costs $17, which corresponds to choice B) $17.