Question

the function ( c(t)=15t + 120 ) models the total cost of a gym membership, where ( t ) is the number of months. what is the meaning of the ( y )-intercept in this context?
the initial sign - up fee for the gym membership.
the recurring monthly cost of the gym membership.
the number of months required for a specific cost.
the total membership cost accumulated after one year.

Explanation:

The function \(C(t)=15t + 120\) is in the slope - intercept form \(y = mx + b\), where \(m\) is the slope and \(b\) is the \(y\) - intercept. In the context of a cost function \(C(t)\) (where \(t\) is the number of months), when \(t = 0\) (i.e., at the start, before any months of membership have passed), \(C(0)=15\times0+120=120\). This initial cost (\(t = 0\)) represents the amount one has to pay just to sign up for the gym membership, before any monthly fees are added. The term \(15t\) represents the cost that accumulates over \(t\) months (the recurring monthly cost, since for each month \(t\), the cost increases by 15). The \(y\) - intercept (\(b = 120\)) is not related to the number of months (\(t\) is in the linear term \(15t\)) or the cost after a specific time (like one year, which would require substituting \(t = 12\) into the function \(C(t)\)).

Answer:

The initial sign - up fee for the gym membership.