Question 27·Hard·Linear Functions
A bike-share company charges $9.50 for the first 2 hours of a rental and $4.25 for each additional hour. Which of the following functions gives the total cost , in dollars, of renting a bicycle for hours, where is an integer greater than ?
For price-word-problem functions, first separate the fixed charge from the per-unit (per-hour) charge, then express the number of charged units in terms of the variable (often something like or ). Write the total cost as fixed fee plus (rate × number of charged units), and then compare this structure to the answer choices; if you’re unsure, test a simple value like 3 or 4 in both the model you built from the words and in each option to see which matches.
Hints
Separate the two parts of the pricing
Focus on the two parts of the description: a price for the first 2 hours and a different price for each additional hour. One part should be a fixed number, and one part should involve .
Think about how many hours are charged at $4.25
If you rent the bike for hours in total, how many of those hours are after the first 2 hours? Write that as an expression in terms of .
Combine fixed cost and rate × hours
Write the total cost as: (cost of the first 2 hours) (rate for additional hours) × (number of additional hours). Then see which function matches that structure.
Desmos Guide
Enter the answer choice functions
Type each option into Desmos as a separate function of :
A(h)=4.25h-1B(h)=4.25h+9.50C(h)=9.50+4.25(h-2)D(h)=13.75h-8
Compare values for a specific rental time
Pick a value of greater than 2, such as or . In Desmos, either use a table for each function or directly evaluate each one at that to see the predicted cost from each option.
Match with the real-world calculation
Separately, compute the actual cost from the word problem for your chosen (for example, for 3 hours: $9.50 for the first 2 hours plus $4.25 for 1 additional hour). In Desmos, identify which function gives the same dollar amount; that function corresponds to the correct answer choice.
Step-by-step Explanation
Identify fixed and per-hour parts of the cost
From the problem:
- The first 2 hours always cost $9.50 total. This is a fixed starting cost.
- Each additional hour after those 2 costs $4.25. This is the per-hour rate applied only to hours beyond the first 2.
Write an expression for the number of additional hours
Let be the total number of rental hours, with .
- The first 2 hours are covered by the $9.50.
- The remaining hours are additional hours.
- Number of additional hours total hours first 2 hours .
So the $4.25 rate should be multiplied by , not by .
Build the total cost function and match the choice
Total cost fixed cost for first 2 hours cost for extra hours.
- Fixed cost: $9.50
- Extra-hours cost: $4.25 times the additional hours, or
So the function is
This matches answer choice C.