Question 69·Hard·Linear Functions
Two health clubs, FitCo and StrongGym, charge monthly according to linear models in the number of group classes attended, .
- FitCo: where is the total monthly cost, in dollars.
- StrongGym: a flat monthly fee plus $15 per class.
If the monthly cost is the same at both clubs when , what is the flat monthly fee, in dollars, that StrongGym charges?
For linear cost problems with a flat fee and a per‑unit charge, first write an algebraic expression for each plan, using a variable for any unknown fee. Substitute the given number of units (here, classes) into both expressions, then use the phrase “cost is the same” as your cue to set the two expressions equal. Solve the resulting one‑variable equation carefully, watching your arithmetic, and match that value to the answer choices.
Hints
Start with FitCo’s cost at 4 classes
Use the formula . What do you get if you plug in and simplify?
Write StrongGym’s expression at 4 classes
Let the flat fee be a variable (for example, ). When , what does StrongGym’s cost look like in terms of ?
Use the fact that the costs are the same
Set the cost expressions you found for equal to each other and solve the resulting equation for the flat fee.
Desmos Guide
Compute the total shared cost at 4 classes
In Desmos, type 38 + 12*4 on one line. The value that appears is the common total cost when 4 classes are taken at both clubs.
Isolate StrongGym’s flat fee numerically
On a new line, type 38 + 12*4 - 15*4. The result shown is the flat monthly fee StrongGym must charge so that both clubs have the same cost at 4 classes.
Step-by-step Explanation
Write an expression for StrongGym
FitCo’s cost is given by
For StrongGym, the problem says its cost is a flat monthly fee plus $15 per class. Let the flat fee be dollars. Then StrongGym’s cost model is
Find FitCo’s cost when
Substitute into FitCo’s formula:
Calculate and then add 38 to find FitCo’s total cost at 4 classes. Keep that number in mind; StrongGym’s cost at 4 classes must match it.
Write StrongGym’s cost when
Substitute into StrongGym’s model:
Simplify :
This is StrongGym’s total monthly cost when a member takes 4 classes.
Set the costs equal and solve for the flat fee
The problem says the monthly cost is the same at both clubs when , so set the two expressions equal:
From the earlier steps, this gives
Compute the left side:
Solve for by subtracting 60 from both sides:
So the flat monthly fee StrongGym charges is $26, which corresponds to choice B) 26.