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?
A linear model adds a fixed fee to a charge that repeats for each class. When two clubs cost the same at a given number of classes, evaluate the known model in Desmos and set the other club’s total equal to it. Keep the total separate from the flat fee: the question may ask for only one part of the cost.
Hints
- Hint 1
In a linear model, the per-class charge is multiplied by the number of classes, but the flat fee is paid once. If is StrongGym’s flat fee, how would you write its cost for four classes?
- Hint 2
Evaluate FitCo’s total at four classes by entering its function in Desmos and then typing . StrongGym must match that total, but the total isn’t the flat fee the question asks for.
- Hint 3
The costs being the same at means StrongGym’s fee plus four class charges equals FitCo’s four-class cost. Set those totals equal, then use in Desmos to find the fee.
Step-by-step
Match the four-class totals
Step 1Write StrongGym’s four-class cost
Let be StrongGym’s flat fee, in dollars. Its four classes cost , so its total is . The flat fee is paid once; only the per-class charge gets multiplied by the number of classes.
- Step 2
Find FitCo’s four-class total
Type FitCo’s given function in Desmos, then type . Desmos shows . That is FitCo’s total cost for four classes, not StrongGym’s flat fee.
- Step 3
Set the totals equal
The clubs charge the same total at four classes, so StrongGym’s fee plus its class charges must equal FitCo’s : . Don’t set the per-class charges equal; the condition is about the whole monthly bill.
- Step 4
Find the flat fee
Type in Desmos. The asks Desmos to find ; under PARAMETERS, it shows . Since is StrongGym’s flat monthly fee, StrongGym charges $26 before any class charges. Choice B.