To compare the rates of change for both DVD rental stores, we first need to analyze the costs at each store.
First Store:
The cost data for the first store is as follows:
- 1 DVD: $5.50
- 2 DVDs: $7.00
- 3 DVDs: $8.50
We can calculate the change in cost as the number of DVDs increases.
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From 1 to 2 DVDs: \[ \text{Change in cost} = 7.00 - 5.50 = 1.50 \] \[ \text{Change in DVDs} = 2 - 1 = 1 \] \[ \text{Rate of change} = \frac{1.50}{1} = 1.50 \text{ per DVD} \]
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From 2 to 3 DVDs: \[ \text{Change in cost} = 8.50 - 7.00 = 1.50 \] \[ \text{Change in DVDs} = 3 - 2 = 1 \] \[ \text{Rate of change} = \frac{1.50}{1} = 1.50 \text{ per DVD} \]
Thus, the rate of change for the first store is constant at $1.50 per DVD rented.
Second Store:
The cost for the second store is modeled by the function: \[ C = 5 + 2d \]
This is a linear equation where:
- The fixed membership fee is $5.
- The fee per DVD rented is $2.
The rate of change, or the slope of the linear function, is given by the coefficient of \(d\), which is 2. This means that for each DVD rented, the cost increases by $2.
Comparison of Rates of Change:
- First Store Rate of Change: $1.50 per DVD
- Second Store Rate of Change: $2.00 per DVD
Conclusion about the Context:
- The first store has a lower rate of change in cost ($1.50) compared to the second store ($2.00).
- This means that renting DVDs from the first store is less expensive per DVD compared to the second store, at least for the number of DVDs rented shown in the data.
- Therefore, if Donnie plans to rent multiple DVDs, the first store would be more economical than the second store based on the per DVD rental cost.
As a general recommendation based on the current rental rates, Donnie might consider renting DVDs from the first store to save money on rentals.
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