Question
The cost, in dollars, to produce x designer dog leashes is C(x)=10x+8, and the revenue function, in dollars, is R(x)= -3x^2+94x
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 \( R(x) \):
### Step 1: Find the Profit Function
The profit function \( P(x) \) is given by:
\[
P(x) = R(x) - C(x)
\]
Substituting in the revenue and cost functions:
\[
P(x) = (-3x^2 + 94x) - (10x + 8)
\]
Simplifying this, we combine like terms:
\[
P(x) = -3x^2 + 94x - 10x - 8
\]
\[
P(x) = -3x^2 + 84x - 8
\]
### Step 2: Find the number of leashes which need to be sold to maximize profit
The profit function \( P(x) = -3x^2 + 84x - 8 \) is a quadratic function that opens downwards (since the coefficient of \( x^2 \) is negative). To find the maximum profit, we can use the vertex formula for a quadratic, \( x = -\frac{b}{2a} \), where \( a = -3 \) and \( b = 84 \):
\[
x = -\frac{84}{2 \cdot -3} = \frac{84}{6} = 14
\]
So, **14 leashes need to be sold to maximize the profit**.
### Step 3: Find the maximum profit
To find the maximum profit, we substitute \( x = 14 \) back into the profit function:
\[
P(14) = -3(14^2) + 84(14) - 8
\]
Calculating \( 14^2 \):
\[
14^2 = 196
\]
Now substitute:
\[
P(14) = -3(196) + 84(14) - 8
\]
\[
= -588 + 1176 - 8
\]
\[
= 580
\]
So, **the maximum profit is $580**.
### Step 4: Find the price to charge per leash to maximize profit
To find the price per leash, we need to find the revenue when 14 leashes are sold. The revenue function is given by:
\[
R(x) = -3x^2 + 94x
\]
Substituting in \( x = 14 \):
\[
R(14) = -3(14^2) + 94(14)
\]
Continuing from our earlier calculation of \( 14^2 \):
\[
R(14) = -588 + 1316
\]
\[
= 728
\]
Now, the price per leash \( p \) is:
\[
p = \frac{R(14)}{14} = \frac{728}{14} = 52
\]
So, **the price to charge per leash to maximize profit is $52**.
### Summary:
- Profit function: \( P(x) = -3x^2 + 84x - 8 \)
- Number of leashes to maximize profit: **14 leashes**
- Maximum profit: **$580**
- Price per leash to maximize profit: **$52**
### Reasons to Pay or Not Pay $52:
**Reasons to Pay:**
1. **Quality and Durability:** If the leashes are of high quality and durability, a higher price might be justified.
2. **Brand Value:** A designer brand might offer a status symbol that attracts buyers.
3. **Special Features:** If the leashes have unique features (e.g., reflective material, eco-friendly materials), consumers may be willing to pay more.
**Reasons Not to Pay:**
1. **Budget Constraints:** Consumers may have limited budgets that prevent them from purchasing higher-cost items.
2. **Comparative Pricing:** If similar products are available at lower prices, consumers may choose alternatives.
3. **Assessment of Value:** If consumers do not perceive the additional features or quality to justify the cost, they may opt not to buy.
### Step 1: Find the Profit Function
The profit function \( P(x) \) is given by:
\[
P(x) = R(x) - C(x)
\]
Substituting in the revenue and cost functions:
\[
P(x) = (-3x^2 + 94x) - (10x + 8)
\]
Simplifying this, we combine like terms:
\[
P(x) = -3x^2 + 94x - 10x - 8
\]
\[
P(x) = -3x^2 + 84x - 8
\]
### Step 2: Find the number of leashes which need to be sold to maximize profit
The profit function \( P(x) = -3x^2 + 84x - 8 \) is a quadratic function that opens downwards (since the coefficient of \( x^2 \) is negative). To find the maximum profit, we can use the vertex formula for a quadratic, \( x = -\frac{b}{2a} \), where \( a = -3 \) and \( b = 84 \):
\[
x = -\frac{84}{2 \cdot -3} = \frac{84}{6} = 14
\]
So, **14 leashes need to be sold to maximize the profit**.
### Step 3: Find the maximum profit
To find the maximum profit, we substitute \( x = 14 \) back into the profit function:
\[
P(14) = -3(14^2) + 84(14) - 8
\]
Calculating \( 14^2 \):
\[
14^2 = 196
\]
Now substitute:
\[
P(14) = -3(196) + 84(14) - 8
\]
\[
= -588 + 1176 - 8
\]
\[
= 580
\]
So, **the maximum profit is $580**.
### Step 4: Find the price to charge per leash to maximize profit
To find the price per leash, we need to find the revenue when 14 leashes are sold. The revenue function is given by:
\[
R(x) = -3x^2 + 94x
\]
Substituting in \( x = 14 \):
\[
R(14) = -3(14^2) + 94(14)
\]
Continuing from our earlier calculation of \( 14^2 \):
\[
R(14) = -588 + 1316
\]
\[
= 728
\]
Now, the price per leash \( p \) is:
\[
p = \frac{R(14)}{14} = \frac{728}{14} = 52
\]
So, **the price to charge per leash to maximize profit is $52**.
### Summary:
- Profit function: \( P(x) = -3x^2 + 84x - 8 \)
- Number of leashes to maximize profit: **14 leashes**
- Maximum profit: **$580**
- Price per leash to maximize profit: **$52**
### Reasons to Pay or Not Pay $52:
**Reasons to Pay:**
1. **Quality and Durability:** If the leashes are of high quality and durability, a higher price might be justified.
2. **Brand Value:** A designer brand might offer a status symbol that attracts buyers.
3. **Special Features:** If the leashes have unique features (e.g., reflective material, eco-friendly materials), consumers may be willing to pay more.
**Reasons Not to Pay:**
1. **Budget Constraints:** Consumers may have limited budgets that prevent them from purchasing higher-cost items.
2. **Comparative Pricing:** If similar products are available at lower prices, consumers may choose alternatives.
3. **Assessment of Value:** If consumers do not perceive the additional features or quality to justify the cost, they may opt not to buy.