To solve this problem, we need to follow several steps.
Step 1: Finding the Profit Function
The profit function \( P(x) \) is calculated as the difference between revenue \( R(x) \) and cost \( C(x) \).
Given:
- Revenue function: \( R(x) = -4x^2 + 92x \)
- Cost function: \( C(x) = 4x + 9 \)
The profit function is: \[ P(x) = R(x) - C(x) \] Substituting the functions: \[ P(x) = (-4x^2 + 92x) - (4x + 9) \] Simplifying this gives: \[ P(x) = -4x^2 + 92x - 4x - 9 = -4x^2 + 88x - 9 \]
Step 2: Finding the Number of Leashes to Maximize Profit
The profit function \( P(x) = -4x^2 + 88x - 9 \) is a quadratic equation. To find the maximum profit, we calculate the vertex of the parabola defined by this function. The x-coordinate of the vertex of a quadratic \( ax^2 + bx + c \) is given by the formula: \[ x = -\frac{b}{2a} \] Here, \( a = -4 \) and \( b = 88 \): \[ x = -\frac{88}{2(-4)} = \frac{88}{8} = 11 \]
Thus, the number of leashes which need to be sold to maximize the profit is: 11 leashes.
Step 3: Finding the Maximum Profit
To find the maximum profit, substitute \( x = 11 \) into the profit function \( P(x) \): \[ P(11) = -4(11)^2 + 88(11) - 9 \] Calculating this: \[ P(11) = -4(121) + 968 - 9 = -484 + 968 - 9 = 475 \]
So the maximum profit is: $475.
Step 4: Finding the Price to Charge per Leash
To find the price to charge per leash to maximize profit, we need to find the revenue when selling 11 leashes: \[ R(11) = -4(11)^2 + 92(11) \] Calculating this: \[ R(11) = -4(121) + 1012 = -484 + 1012 = 528 \] The total revenue from selling 11 leashes is $528. To find the price per leash, we divide the revenue by the number of leashes: \[ \text{Price per leash} = \frac{R(11)}{11} = \frac{528}{11} = 48 \]
The price to charge per leash to maximize profit is: $48.
Summary
- Profit Function: \( P(x) = -4x^2 + 88x - 9 \)
- Number of leashes to maximize profit: 11 leashes
- Maximum profit: $475
- Price per leash to maximize profit: $48
Best Reasons to Either Pay or Not Pay That Much for a Leash
Reasons to Pay for a Designer Leash:
- Quality Materials: Designer leashes are typically made from high-quality materials that ensure durability and longevity.
- Design and Aesthetic Appeal: People may prefer a leash that looks stylish and complements their dog's appearance or their own personal style.
- Functionality: Designer leashes often come with added features such as ergonomic handles, safety components, or custom fittings that improve the experience for both owner and dog.
- Status Symbol: Some consumers buy designer items as a representation of status or lifestyle.
Reasons Not to Pay That Much:
- Functionality vs. Price: Many leashes serve the same basic function at a much lower price, making it hard to justify the premium.
- Affordability: Not everyone can afford designer prices, and there may be adequate alternatives.
- Simplicity: For some pet owners, a leash does not need to be designer; functionality is more important than appearance.
- Trends: Trends change, and what is considered "fashionable" might not match the owner's style in the future.
Consumers need to weigh the importance of these factors when deciding whether to invest in a designer product.