Question
The cost, in dollars, to produce x designer dog leashes is C(x)=5x+4, and the revenue function, in dollars, is R(x)= -3x^2+53x
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 solve the problem, we start by finding the profit function \( P(x) \). The profit \( P(x) \) is calculated as the difference between the revenue \( R(x) \) and the cost \( C(x) \):
\[
P(x) = R(x) - C(x)
\]
Given:
- Cost function: \( C(x) = 5x + 4 \)
- Revenue function: \( R(x) = -3x^2 + 53x \)
Now, we can calculate the profit function:
\[
P(x) = (-3x^2 + 53x) - (5x + 4)
\]
Simplifying this further:
\[
P(x) = -3x^2 + 53x - 5x - 4
\]
\[
P(x) = -3x^2 + 48x - 4
\]
Now we have the profit function:
\[
P(x) = -3x^2 + 48x - 4
\]
### Finding the number of leashes to maximize profit
To find the number of leashes that need to be sold to maximize profit, we can use the vertex formula for a quadratic function \( ax^2 + bx + c \). The x-coordinate of the vertex (which gives the maximum point for a downward-opening parabola) is given by:
\[
x = -\frac{b}{2a}
\]
For our profit function:
- \( a = -3 \)
- \( b = 48 \)
Now we can calculate:
\[
x = -\frac{48}{2(-3)} = \frac{48}{6} = 8
\]
So, **the number of leashes which need to be sold to maximize the profit is 8 leashes**.
### Finding the maximum profit
To find the maximum profit, we substitute \( x = 8 \) back into the profit function:
\[
P(8) = -3(8^2) + 48(8) - 4
\]
\[
P(8) = -3(64) + 384 - 4
\]
\[
P(8) = -192 + 384 - 4
\]
\[
P(8) = 188
\]
So, **the maximum profit is 188 dollars**.
### Finding the price to charge per leash to maximize profit
To find the price per leash to maximize profit, we first need to calculate the revenue when 8 leashes are sold. We substitute back into the revenue function:
\[
R(8) = -3(8^2) + 53(8)
\]
\[
R(8) = -3(64) + 424
\]
\[
R(8) = -192 + 424
\]
\[
R(8) = 232
\]
Now, we find the price per leash:
\[
\text{Price per leash} = \frac{R(8)}{8} = \frac{232}{8} = 29
\]
So, **the price to charge per leash to maximize profit is 29 dollars**.
### Summary
- **Number of leashes to maximize profit:** 8 leashes
- **Maximum profit:** 188 dollars
- **Price to charge per leash to maximize profit:** 29 dollars
### Best reasons to either pay or not pay that much for a leash:
**Reasons to pay:**
1. **Quality Assurance:** Designer dog leashes are typically made from high-quality materials, ensuring durability and safety.
2. **Style and Design:** Unique and stylish designs may enhance the owner's experience and satisfy their aesthetic preferences.
3. **Functionality:** A well-designed leash may offer better functionality, such as comfort for the handler and safety features for the dog.
**Reasons not to pay:**
1. **Cost vs. Utility:** Some may feel that the price doesn't correspond to the necessary utility, especially if less expensive options are available.
2. **Economic Constraints:** If a buyer has limited finances, they may opt for more affordable alternatives without significantly sacrificing quality.
3. **Alternatives Available:** Many alternatives in the market possibly provide similar functionality at a lower price.
These considerations depend significantly on individual priorities, lifestyles, and the financial situation of potential buyers.
\[
P(x) = R(x) - C(x)
\]
Given:
- Cost function: \( C(x) = 5x + 4 \)
- Revenue function: \( R(x) = -3x^2 + 53x \)
Now, we can calculate the profit function:
\[
P(x) = (-3x^2 + 53x) - (5x + 4)
\]
Simplifying this further:
\[
P(x) = -3x^2 + 53x - 5x - 4
\]
\[
P(x) = -3x^2 + 48x - 4
\]
Now we have the profit function:
\[
P(x) = -3x^2 + 48x - 4
\]
### Finding the number of leashes to maximize profit
To find the number of leashes that need to be sold to maximize profit, we can use the vertex formula for a quadratic function \( ax^2 + bx + c \). The x-coordinate of the vertex (which gives the maximum point for a downward-opening parabola) is given by:
\[
x = -\frac{b}{2a}
\]
For our profit function:
- \( a = -3 \)
- \( b = 48 \)
Now we can calculate:
\[
x = -\frac{48}{2(-3)} = \frac{48}{6} = 8
\]
So, **the number of leashes which need to be sold to maximize the profit is 8 leashes**.
### Finding the maximum profit
To find the maximum profit, we substitute \( x = 8 \) back into the profit function:
\[
P(8) = -3(8^2) + 48(8) - 4
\]
\[
P(8) = -3(64) + 384 - 4
\]
\[
P(8) = -192 + 384 - 4
\]
\[
P(8) = 188
\]
So, **the maximum profit is 188 dollars**.
### Finding the price to charge per leash to maximize profit
To find the price per leash to maximize profit, we first need to calculate the revenue when 8 leashes are sold. We substitute back into the revenue function:
\[
R(8) = -3(8^2) + 53(8)
\]
\[
R(8) = -3(64) + 424
\]
\[
R(8) = -192 + 424
\]
\[
R(8) = 232
\]
Now, we find the price per leash:
\[
\text{Price per leash} = \frac{R(8)}{8} = \frac{232}{8} = 29
\]
So, **the price to charge per leash to maximize profit is 29 dollars**.
### Summary
- **Number of leashes to maximize profit:** 8 leashes
- **Maximum profit:** 188 dollars
- **Price to charge per leash to maximize profit:** 29 dollars
### Best reasons to either pay or not pay that much for a leash:
**Reasons to pay:**
1. **Quality Assurance:** Designer dog leashes are typically made from high-quality materials, ensuring durability and safety.
2. **Style and Design:** Unique and stylish designs may enhance the owner's experience and satisfy their aesthetic preferences.
3. **Functionality:** A well-designed leash may offer better functionality, such as comfort for the handler and safety features for the dog.
**Reasons not to pay:**
1. **Cost vs. Utility:** Some may feel that the price doesn't correspond to the necessary utility, especially if less expensive options are available.
2. **Economic Constraints:** If a buyer has limited finances, they may opt for more affordable alternatives without significantly sacrificing quality.
3. **Alternatives Available:** Many alternatives in the market possibly provide similar functionality at a lower price.
These considerations depend significantly on individual priorities, lifestyles, and the financial situation of potential buyers.