Question
The cost, in dollars, to produce x designer dog leashes is C(x)= 10x +6, and the revenue function, in dollars, is R(x)=-4x^2+114x
Find the profit function.
P(x)=
Find the number of leashes which need to be sold to maximize the profit.
leashes
dogs
dollars
Find the maximum profit.
leashes
dogs
dollars
Find the price to charge per leash to maximize profit.
dogs
leashes
dollars
What would be the best reasons to either pay or not pay that much for a leash?
Find the profit function.
P(x)=
Find the number of leashes which need to be sold to maximize the profit.
leashes
dogs
dollars
Find the maximum profit.
leashes
dogs
dollars
Find the price to charge per leash to maximize profit.
dogs
leashes
dollars
What would be the best reasons to either pay or not pay that much for a leash?
Answers
GPT-4o mini
To find the profit function, we can subtract the cost function \(C(x)\) from the revenue function \(R(x)\):
### Profit Function
\[
P(x) = R(x) - C(x)
\]
\[
P(x) = (-4x^2 + 114x) - (10x + 6)
\]
\[
P(x) = -4x^2 + 114x - 10x - 6
\]
\[
P(x) = -4x^2 + 104x - 6
\]
### Maximizing Profit
To find the number of leashes \(x\) which maximizes profit, we need to locate the vertex of the quadratic function \(P(x) = -4x^2 + 104x - 6\). The vertex occurs at:
\[
x = -\frac{b}{2a}
\]
where \(a = -4\) and \(b = 104\).
\[
x = -\frac{104}{2 \cdot -4} = \frac{104}{8} = 13
\]
### Maximum Profit
Now, substitute \(x = 13\) back into the profit function to find the maximum profit:
\[
P(13) = -4(13)^2 + 104(13) - 6
\]
\[
= -4(169) + 1352 - 6
\]
\[
= -676 + 1352 - 6
\]
\[
= 670
\]
### Price Per Leash
To find the price to charge per leash that maximizes profit, substitute \(x = 13\) into the revenue function:
\[
R(13) = -4(13^2) + 114(13)
\]
\[
= -4(169) + 1482
\]
\[
= -676 + 1482
\]
\[
= 806
\]
Then, since \(R(x)\) is the total revenue, we divide by \(x\):
\[
\text{Price per leash} = \frac{R(13)}{13} = \frac{806}{13} \approx 62.00
\]
### Summary of Results
1. **Profit Function**: \(P(x) = -4x^2 + 104x - 6\)
2. **Number of leashes to maximize profit**: \(13\) leashes
3. **Maximum profit**: \(670\) dollars
4. **Price to charge per leash to maximize profit**: approximately \(62.00\) dollars
### Reasoning for Charging That Price
**Reasons to Pay:**
- **Quality Craftsmanship**: If the leash is made of durable materials, it can justify the higher price.
- **Brand Value**: If the designer label holds significant value or prestige, customers might be willing to pay more.
- **Unique Features**: Features such as retractable mechanisms, customizable designs, or eco-friendly materials may appeal to customers.
**Reasons Not to Pay:**
- **Availability of Alternatives**: If there are cheaper, yet comparable options available, consumers may opt for those.
- **Cost Sensitivity**: Some customers may have a limited budget for pet supplies, making it difficult to justify higher prices.
- **Perceived Value vs. Actual Utility**: If consumers do not perceive a difference in quality or usability compared to cheaper options, they may reject high pricing.
In summary, while a $62 leash may attract specific customers willing to pay for quality, others might resist based on budget constraints and availability of lower-priced alternatives.
### Profit Function
\[
P(x) = R(x) - C(x)
\]
\[
P(x) = (-4x^2 + 114x) - (10x + 6)
\]
\[
P(x) = -4x^2 + 114x - 10x - 6
\]
\[
P(x) = -4x^2 + 104x - 6
\]
### Maximizing Profit
To find the number of leashes \(x\) which maximizes profit, we need to locate the vertex of the quadratic function \(P(x) = -4x^2 + 104x - 6\). The vertex occurs at:
\[
x = -\frac{b}{2a}
\]
where \(a = -4\) and \(b = 104\).
\[
x = -\frac{104}{2 \cdot -4} = \frac{104}{8} = 13
\]
### Maximum Profit
Now, substitute \(x = 13\) back into the profit function to find the maximum profit:
\[
P(13) = -4(13)^2 + 104(13) - 6
\]
\[
= -4(169) + 1352 - 6
\]
\[
= -676 + 1352 - 6
\]
\[
= 670
\]
### Price Per Leash
To find the price to charge per leash that maximizes profit, substitute \(x = 13\) into the revenue function:
\[
R(13) = -4(13^2) + 114(13)
\]
\[
= -4(169) + 1482
\]
\[
= -676 + 1482
\]
\[
= 806
\]
Then, since \(R(x)\) is the total revenue, we divide by \(x\):
\[
\text{Price per leash} = \frac{R(13)}{13} = \frac{806}{13} \approx 62.00
\]
### Summary of Results
1. **Profit Function**: \(P(x) = -4x^2 + 104x - 6\)
2. **Number of leashes to maximize profit**: \(13\) leashes
3. **Maximum profit**: \(670\) dollars
4. **Price to charge per leash to maximize profit**: approximately \(62.00\) dollars
### Reasoning for Charging That Price
**Reasons to Pay:**
- **Quality Craftsmanship**: If the leash is made of durable materials, it can justify the higher price.
- **Brand Value**: If the designer label holds significant value or prestige, customers might be willing to pay more.
- **Unique Features**: Features such as retractable mechanisms, customizable designs, or eco-friendly materials may appeal to customers.
**Reasons Not to Pay:**
- **Availability of Alternatives**: If there are cheaper, yet comparable options available, consumers may opt for those.
- **Cost Sensitivity**: Some customers may have a limited budget for pet supplies, making it difficult to justify higher prices.
- **Perceived Value vs. Actual Utility**: If consumers do not perceive a difference in quality or usability compared to cheaper options, they may reject high pricing.
In summary, while a $62 leash may attract specific customers willing to pay for quality, others might resist based on budget constraints and availability of lower-priced alternatives.