Question
The cost, in dollars, to produce x designer dog leashes is C(x)=5x+7, and the revenue function, in dollars, is R(x)= -4x^2+149x
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 use the formula for profit:
\[ P(x) = R(x) - C(x) \]
Where \( R(x) \) is the revenue function and \( C(x) \) is the cost function.
### Step 1: Find the Profit Function
Given:
- \( C(x) = 5x + 7 \)
- \( R(x) = -4x^2 + 149x \)
The profit function will be:
\[
P(x) = R(x) - C(x)
\]
Substituting the values:
\[
P(x) = (-4x^2 + 149x) - (5x + 7)
\]
\[
P(x) = -4x^2 + 149x - 5x - 7
\]
\[
P(x) = -4x^2 + 144x - 7
\]
### Step 2: Find the Number of Leashes that Maximizes Profit
To find the maximum profit, we need to find the vertex of the quadratic function \( P(x) = -4x^2 + 144x - 7 \). The x-coordinate of the vertex is given by the formula:
\[
x = -\frac{b}{2a}
\]
where \( a = -4 \) and \( b = 144 \).
Calculating:
\[
x = -\frac{144}{2 \cdot -4} = \frac{144}{8} = 18
\]
So, the number of leashes that need to be sold to maximize profit is **18 leashes**.
### Step 3: Find the Maximum Profit
To find the maximum profit, substitute \( x = 18 \) back into the profit function \( P(x) \):
\[
P(18) = -4(18^2) + 144(18) - 7
\]
Calculating \( 18^2 = 324 \):
\[
P(18) = -4(324) + 144(18) - 7
\]
\[
P(18) = -1296 + 2592 - 7
\]
\[
P(18) = 1296 - 7 = 1289
\]
So, the maximum profit is **$1289**.
### Step 4: Find the Price to Charge per Leash to Maximize Profit
The price per leash is given by the revenue function. To find the selling price per leash at \( x = 18 \):
\[
R(18) = -4(18^2) + 149(18)
\]
\[
R(18) = -4(324) + 149(18)
\]
Calculating:
\[
R(18) = -1296 + 2682
\]
\[
R(18) = 1386
\]
To find the price per leash:
\[
\text{Price per leash} = \frac{R(18)}{18} = \frac{1386}{18} = 77
\]
So, the price to charge per leash to maximize profit is **$77**.
### Summary:
- The profit function is:
\[
P(x) = -4x^2 + 144x - 7
\]
- The number of leashes to sell to maximize profit: **18 leashes**
- The maximum profit: **$1289**
- The price to charge per leash to maximize profit: **$77**
### Step 5: Best Reasons to Pay or Not Pay That Much for a Leash
**Reasons to Pay $77:**
1. **Quality Materials:** Designer leashes often use higher-quality materials that enhance durability and style.
2. **Brand Reputation:** A designer leash can come with brand prestige, reflecting status and style.
3. **Unique Features:** These leashes may have unique designs, safety features, or added comfort for the dog.
4. **Enhanced Customer Experience:** Buying from a designer brand often includes better customer service and shopping experience.
**Reasons Not to Pay $77:**
1. **Affordability:** Some pet owners may not be able to justify or afford the high price, especially if they have multiple pets.
2. **Functionality vs. Cost:** Many functional leashes can achieve the same result at a much lower price.
3. **Trends Change:** Fashion trends in pet accessories can change, potentially making a designer item less appealing over time.
4. **Value Over Time:** Depending on a pet's behavior, a leash may quickly wear out or become obsolete, questioning the investment.
These factors can affect a consumer's decision to purchase a designer dog leash at the higher price point of $77.
\[ P(x) = R(x) - C(x) \]
Where \( R(x) \) is the revenue function and \( C(x) \) is the cost function.
### Step 1: Find the Profit Function
Given:
- \( C(x) = 5x + 7 \)
- \( R(x) = -4x^2 + 149x \)
The profit function will be:
\[
P(x) = R(x) - C(x)
\]
Substituting the values:
\[
P(x) = (-4x^2 + 149x) - (5x + 7)
\]
\[
P(x) = -4x^2 + 149x - 5x - 7
\]
\[
P(x) = -4x^2 + 144x - 7
\]
### Step 2: Find the Number of Leashes that Maximizes Profit
To find the maximum profit, we need to find the vertex of the quadratic function \( P(x) = -4x^2 + 144x - 7 \). The x-coordinate of the vertex is given by the formula:
\[
x = -\frac{b}{2a}
\]
where \( a = -4 \) and \( b = 144 \).
Calculating:
\[
x = -\frac{144}{2 \cdot -4} = \frac{144}{8} = 18
\]
So, the number of leashes that need to be sold to maximize profit is **18 leashes**.
### Step 3: Find the Maximum Profit
To find the maximum profit, substitute \( x = 18 \) back into the profit function \( P(x) \):
\[
P(18) = -4(18^2) + 144(18) - 7
\]
Calculating \( 18^2 = 324 \):
\[
P(18) = -4(324) + 144(18) - 7
\]
\[
P(18) = -1296 + 2592 - 7
\]
\[
P(18) = 1296 - 7 = 1289
\]
So, the maximum profit is **$1289**.
### Step 4: Find the Price to Charge per Leash to Maximize Profit
The price per leash is given by the revenue function. To find the selling price per leash at \( x = 18 \):
\[
R(18) = -4(18^2) + 149(18)
\]
\[
R(18) = -4(324) + 149(18)
\]
Calculating:
\[
R(18) = -1296 + 2682
\]
\[
R(18) = 1386
\]
To find the price per leash:
\[
\text{Price per leash} = \frac{R(18)}{18} = \frac{1386}{18} = 77
\]
So, the price to charge per leash to maximize profit is **$77**.
### Summary:
- The profit function is:
\[
P(x) = -4x^2 + 144x - 7
\]
- The number of leashes to sell to maximize profit: **18 leashes**
- The maximum profit: **$1289**
- The price to charge per leash to maximize profit: **$77**
### Step 5: Best Reasons to Pay or Not Pay That Much for a Leash
**Reasons to Pay $77:**
1. **Quality Materials:** Designer leashes often use higher-quality materials that enhance durability and style.
2. **Brand Reputation:** A designer leash can come with brand prestige, reflecting status and style.
3. **Unique Features:** These leashes may have unique designs, safety features, or added comfort for the dog.
4. **Enhanced Customer Experience:** Buying from a designer brand often includes better customer service and shopping experience.
**Reasons Not to Pay $77:**
1. **Affordability:** Some pet owners may not be able to justify or afford the high price, especially if they have multiple pets.
2. **Functionality vs. Cost:** Many functional leashes can achieve the same result at a much lower price.
3. **Trends Change:** Fashion trends in pet accessories can change, potentially making a designer item less appealing over time.
4. **Value Over Time:** Depending on a pet's behavior, a leash may quickly wear out or become obsolete, questioning the investment.
These factors can affect a consumer's decision to purchase a designer dog leash at the higher price point of $77.