Question
Donnie is considering two different DVD rental stores, both of which have a one-time membership fee and a fee per rented DVD.
The cost for renting DVDs at the first store is shown below:
Number of DVDs rentedCost ($)15.5027.0038.50
The cost for renting DVDs at the second store is modeled by the linear function C = 5 + 2d, where d is the number of DVDs rented and C is the cost.
Compare the rates of change for each function and explain what this means in terms of the context.
The cost for renting DVDs at the first store is shown below:
Number of DVDs rentedCost ($)15.5027.0038.50
The cost for renting DVDs at the second store is modeled by the linear function C = 5 + 2d, where d is the number of DVDs rented and C is the cost.
Compare the rates of change for each function and explain what this means in terms of the context.
Answers
GPT-4o mini
To compare the rates of change of the costs for renting DVDs at the two stores, we need to analyze the costs provided for the first store and the linear function for the second store.
### Cost Analysis:
1. **First Store:**
- Basing on the costs given:
- Renting 1 DVD: $15.50
- Renting 2 DVDs: $27.00
- Renting 3 DVDs: $38.50
We can calculate the cost per additional DVD rented:
- From 1 to 2 DVDs:
\[
\text{Cost difference} = 27.00 - 15.50 = 11.50
\]
- From 2 to 3 DVDs:
\[
\text{Cost difference} = 38.50 - 27.00 = 11.50
\]
Here, the cost per DVD increases since it jumps from a total of $15.50 to $27.00 (costing $11.50 for the second DVD) and from $27.00 to $38.50 (again costing $11.50 for the third DVD). However, it seems consistent for the few transactions we have; we need more data points to check for any variations in the rate of change since the given amounts only cover three rentals.
2. **Second Store:**
- The cost function is given by \( C = 5 + 2d \), where:
- The membership fee is $5.
- The cost per additional DVD (\( d \)) is $2.
The rate of change here is consistent: for every additional DVD rented, the cost increases by $2.
### Comparison of Rates of Change:
- **First Store:**
- The cost per rented DVD appears to be a fixed increase of $11.50 as observed from the provided data, but this is only based on the three data points.
- **Second Store:**
- The cost increases consistently and linearly by $2 for each DVD rented beyond the membership fee.
### Explanation in Terms of Context:
1. **Cost Efficiency:**
- The second store offers a lower and more consistent rate of change with a charge of $2 per DVD rented. This is advantageous for customers who rent multiple DVDs, as the costs do not exponentially increase.
2. **First Store Pricing:**
- The first store's pricing appears less predictable; with only three data points, one might assume it is static based on those ranges, but typically, it would likely involve diminishing or increasing returns in rent pricing or include other variable costs. However, without more data, we can't clearly define their model or rule out their being non-linear in pricing.
3. **Decision Making:**
- Based on these observations, if Donnie plans to rent multiple DVDs, the second store would be a more cost-effective choice because of the clearly defined and lower marginal cost per DVD. If he only plans to rent one or two DVDs, then the first store might be worth considering for comparison, though the initial higher price could be prohibitive.
In conclusion, for frequent rentals, the second store is financially advantageous, while the first store remains uncertain and potentially more expensive, particularly for multiple rentals.
### Cost Analysis:
1. **First Store:**
- Basing on the costs given:
- Renting 1 DVD: $15.50
- Renting 2 DVDs: $27.00
- Renting 3 DVDs: $38.50
We can calculate the cost per additional DVD rented:
- From 1 to 2 DVDs:
\[
\text{Cost difference} = 27.00 - 15.50 = 11.50
\]
- From 2 to 3 DVDs:
\[
\text{Cost difference} = 38.50 - 27.00 = 11.50
\]
Here, the cost per DVD increases since it jumps from a total of $15.50 to $27.00 (costing $11.50 for the second DVD) and from $27.00 to $38.50 (again costing $11.50 for the third DVD). However, it seems consistent for the few transactions we have; we need more data points to check for any variations in the rate of change since the given amounts only cover three rentals.
2. **Second Store:**
- The cost function is given by \( C = 5 + 2d \), where:
- The membership fee is $5.
- The cost per additional DVD (\( d \)) is $2.
The rate of change here is consistent: for every additional DVD rented, the cost increases by $2.
### Comparison of Rates of Change:
- **First Store:**
- The cost per rented DVD appears to be a fixed increase of $11.50 as observed from the provided data, but this is only based on the three data points.
- **Second Store:**
- The cost increases consistently and linearly by $2 for each DVD rented beyond the membership fee.
### Explanation in Terms of Context:
1. **Cost Efficiency:**
- The second store offers a lower and more consistent rate of change with a charge of $2 per DVD rented. This is advantageous for customers who rent multiple DVDs, as the costs do not exponentially increase.
2. **First Store Pricing:**
- The first store's pricing appears less predictable; with only three data points, one might assume it is static based on those ranges, but typically, it would likely involve diminishing or increasing returns in rent pricing or include other variable costs. However, without more data, we can't clearly define their model or rule out their being non-linear in pricing.
3. **Decision Making:**
- Based on these observations, if Donnie plans to rent multiple DVDs, the second store would be a more cost-effective choice because of the clearly defined and lower marginal cost per DVD. If he only plans to rent one or two DVDs, then the first store might be worth considering for comparison, though the initial higher price could be prohibitive.
In conclusion, for frequent rentals, the second store is financially advantageous, while the first store remains uncertain and potentially more expensive, particularly for multiple rentals.