Question
The cost, in dollars, to produce x designer dog leashes is C(x)= x+4, and the revenue function, in dollars, is R(x)= -3x^2+73x
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 answer your questions, we will first derive the profit function from the cost and revenue functions provided.
1. **Profit Function:**
The profit function \( P(x) \) can be calculated as the difference between the revenue function \( R(x) \) and the cost function \( C(x) \):
\[
P(x) = R(x) - C(x)
\]
Given:
\[
C(x) = x + 4
\]
\[
R(x) = -3x^2 + 73x
\]
Therefore,
\[
P(x) = (-3x^2 + 73x) - (x + 4)
\]
\[
P(x) = -3x^2 + 73x - x - 4
\]
\[
P(x) = -3x^2 + 72x - 4
\]
Thus, the profit function is:
\[
P(x) = -3x^2 + 72x - 4
\]
2. **Maximizing Profit:**
To find the number of leashes that need to be sold to maximize profit, we will find the vertex of the quadratic equation, since the profit function is a downward-opening parabola (the coefficient of \( x^2 \) is negative).
The vertex \( x \) of a quadratic function \( ax^2 + bx + c \) can be found using:
\[
x = -\frac{b}{2a}
\]
Here, \( a = -3 \) and \( b = 72 \):
\[
x = -\frac{72}{2 \times -3} = \frac{72}{6} = 12
\]
Thus, **the number of leashes which need to be sold to maximize profit is 12**.
3. **Maximum Profit:**
Now we substitute \( x = 12 \) into the profit function to find the maximum profit:
\[
P(12) = -3(12^2) + 72(12) - 4
\]
\[
P(12) = -3(144) + 864 - 4
\]
\[
P(12) = -432 + 864 - 4
\]
\[
P(12) = 428
\]
Thus, **the maximum profit is 428 dollars**.
4. **Price to Charge per Leash:**
To find the price to charge per leash to maximize profit, we need to find the revenue when selling 12 leashes:
\[
R(12) = -3(12^2) + 73(12)
\]
\[
R(12) = -3(144) + 876
\]
\[
R(12) = -432 + 876
\]
\[
R(12) = 444
\]
The total revenue from selling 12 leashes is 444 dollars, thus the price per leash is:
\[
\text{Price per leash} = \frac{R(12)}{12} = \frac{444}{12} = 37
\]
Hence, **the price to charge per leash to maximize profit is 37 dollars**.
5. **Reasons to either pay or not pay that much for a leash:**
- **Reasons to Pay:**
- **Quality:** Designer dog leashes are often made with higher-quality materials and can ensure greater durability.
- **Aesthetic Appeal:** They may be more visually appealing and reflect the owner's personality or style.
- **Functionality:** Some designer leashes might include advanced features (like padded grips, reflective materials for safety, etc.) that provide better utility.
- **Brand Recognition:** Owning a designer brand can confer status and perceived value.
- **Reasons Not to Pay:**
- **Affordability:** Paying $37 per leash may not fit into everyone’s budget, especially if multiple leashes are needed.
- **Cost vs. Benefits:** The benefits of a designer leash may not justify the higher cost compared to standard leashes that perform adequately.
- **Wear and Tear:** Leashes can wear out over time, potentially leading to repeated expenses if more expensive options are chosen.
- **Basic Needs:** Many dogs require simply functional leashes, which may be satisfied by lower-cost options.
In summary:
- Profit function: \( P(x) = -3x^2 + 72x - 4 \)
- Leashes to maximize profit: **12**
- Maximum profit: **$428**
- Price per leash: **$37**
- Considerations for purchase: quality vs. cost, function vs. aesthetics, and basic needs versus luxury.
1. **Profit Function:**
The profit function \( P(x) \) can be calculated as the difference between the revenue function \( R(x) \) and the cost function \( C(x) \):
\[
P(x) = R(x) - C(x)
\]
Given:
\[
C(x) = x + 4
\]
\[
R(x) = -3x^2 + 73x
\]
Therefore,
\[
P(x) = (-3x^2 + 73x) - (x + 4)
\]
\[
P(x) = -3x^2 + 73x - x - 4
\]
\[
P(x) = -3x^2 + 72x - 4
\]
Thus, the profit function is:
\[
P(x) = -3x^2 + 72x - 4
\]
2. **Maximizing Profit:**
To find the number of leashes that need to be sold to maximize profit, we will find the vertex of the quadratic equation, since the profit function is a downward-opening parabola (the coefficient of \( x^2 \) is negative).
The vertex \( x \) of a quadratic function \( ax^2 + bx + c \) can be found using:
\[
x = -\frac{b}{2a}
\]
Here, \( a = -3 \) and \( b = 72 \):
\[
x = -\frac{72}{2 \times -3} = \frac{72}{6} = 12
\]
Thus, **the number of leashes which need to be sold to maximize profit is 12**.
3. **Maximum Profit:**
Now we substitute \( x = 12 \) into the profit function to find the maximum profit:
\[
P(12) = -3(12^2) + 72(12) - 4
\]
\[
P(12) = -3(144) + 864 - 4
\]
\[
P(12) = -432 + 864 - 4
\]
\[
P(12) = 428
\]
Thus, **the maximum profit is 428 dollars**.
4. **Price to Charge per Leash:**
To find the price to charge per leash to maximize profit, we need to find the revenue when selling 12 leashes:
\[
R(12) = -3(12^2) + 73(12)
\]
\[
R(12) = -3(144) + 876
\]
\[
R(12) = -432 + 876
\]
\[
R(12) = 444
\]
The total revenue from selling 12 leashes is 444 dollars, thus the price per leash is:
\[
\text{Price per leash} = \frac{R(12)}{12} = \frac{444}{12} = 37
\]
Hence, **the price to charge per leash to maximize profit is 37 dollars**.
5. **Reasons to either pay or not pay that much for a leash:**
- **Reasons to Pay:**
- **Quality:** Designer dog leashes are often made with higher-quality materials and can ensure greater durability.
- **Aesthetic Appeal:** They may be more visually appealing and reflect the owner's personality or style.
- **Functionality:** Some designer leashes might include advanced features (like padded grips, reflective materials for safety, etc.) that provide better utility.
- **Brand Recognition:** Owning a designer brand can confer status and perceived value.
- **Reasons Not to Pay:**
- **Affordability:** Paying $37 per leash may not fit into everyone’s budget, especially if multiple leashes are needed.
- **Cost vs. Benefits:** The benefits of a designer leash may not justify the higher cost compared to standard leashes that perform adequately.
- **Wear and Tear:** Leashes can wear out over time, potentially leading to repeated expenses if more expensive options are chosen.
- **Basic Needs:** Many dogs require simply functional leashes, which may be satisfied by lower-cost options.
In summary:
- Profit function: \( P(x) = -3x^2 + 72x - 4 \)
- Leashes to maximize profit: **12**
- Maximum profit: **$428**
- Price per leash: **$37**
- Considerations for purchase: quality vs. cost, function vs. aesthetics, and basic needs versus luxury.