To compare the rates of change between Alfredo and Alex's gym fees, we first need to calculate the rate of change for both individuals.
Alfredo's Gym Fees
From the table given for Alfredo:
- Month 0: Cost = $20
- Month 1: Cost = $30
- Month 2: Cost = $40
- Month 3: Cost = $50
To find the rate of change, we can look at how much the cost increases per month:
- From Month 0 to Month 1: \(30 - 20 = 10\) (increase in $10)
- From Month 1 to Month 2: \(40 - 30 = 10\) (increase in $10)
- From Month 2 to Month 3: \(50 - 40 = 10\) (increase in $10)
The rate of change for Alfredo is consistently $10 per month.
Alex's Gym Fees
Alex’s cost is given by the equation: \[ c = 25 + 10m \]
From this equation, we see that:
- The fixed cost is $25 (the initial cost when \(m = 0\)),
- The variable cost increases by $10 for every month (the coefficient of \(m\)).
Thus, the rate of change for Alex's fees is also $10 per month.
Comparison
Both Alfredo and Alex have the same rate of change in their gym fees, which is $10 per month. Therefore, neither person has a better rate; they are equal in this regard.
If we consider the total costs after different months:
- At month 0: Alfredo pays $20, Alex pays $25.
- At month 1: Alfredo pays $30, Alex pays $35.
- At month 2: Alfredo pays $40, Alex pays $45.
- At month 3: Alfredo pays $50, Alex pays $55.
While their rates of change are the same, Alfredo starts with a lower initial fee ($20 vs. $25), which could be favorable to him in the short term.