Final answer:
The gym membership cost is based on a linear relationship, represented by the equation T = 20 + 25m, where T is the total cost and m is the number of months. Each month incurs a constant additional cost, contrasting with an exponential relationship where rates change exponentially.
Step-by-step explanation:
The gym membership costing a $20 signing fee plus $25 per month represents a linear relationship between the total cost and the number of months you have the membership. In mathematics, a linear relationship means that as one variable changes, the other variable changes at a constant rate. In this case, each month costs an additional $25 regardless of how long you have the membership, and there's also a one-time signing fee of $20.
For example, the total cost for 3 months would be the signing fee plus 3 times the monthly fee, which calculates to $20 + (3 × $25) = $20 + $75 = $95. This can be expressed with the equation T = 20 + 25m, where T represents the total cost and m represents the number of months.
This contrasts with an exponential relationship, where the rate of change itself increases or decreases exponentially. In this gym membership scenario, the cost does not increase exponentially with each additional month; instead, it increases by the same amount, indicating a linear model.