Question 1197579


**a) Express the price p as a function of the demand x, and find the domain of this function.**

* Given: x = 5,000 - 100p
* Solve for p:
    * 100p = 5,000 - x
    * p = (5,000 - x) / 100
    * p = 50 - 0.01x

* **Domain:** 
    * The demand (x) must be non-negative (you can't sell a negative number of bottles).
    * The price (p) must also be non-negative.
    * 
        * 50 - 0.01x ≥ 0
        * 0.01x ≤ 50
        * x ≤ 5,000

    * **Therefore, the domain of the price function is 0 ≤ x ≤ 5,000**

**b) Find the marginal cost.**

* **Marginal Cost:** The derivative of the cost function with respect to x.
* C(x) = 2,500 + 4x + 0.01x^2
* C'(x) = 4 + 0.02x

**c) Find the revenue function and state its domain.**

* **Revenue (R) = Price (p) * Quantity (x)**
* R(x) = p * x 
* R(x) = (50 - 0.01x) * x 
* R(x) = 50x - 0.01x^2

* **Domain:** The domain of the revenue function is the same as the domain of the price function, which is 0 ≤ x ≤ 5,000.

**d) Find the marginal revenue.**

* **Marginal Revenue:** The derivative of the revenue function with respect to x.
* R(x) = 50x - 0.01x^2
* R'(x) = 50 - 0.02x

**e) Find R′(2,000) and R′(3,000) and interpret these quantities.**

* R'(2,000) = 50 - 0.02 * 2,000 = 50 - 40 = 10
    * When producing 2,000 bottles, the revenue is increasing at a rate of $10 per additional bottle.

* R'(3,000) = 50 - 0.02 * 3,000 = 50 - 60 = -10
    * When producing 3,000 bottles, the revenue is decreasing at a rate of $10 per additional bottle.

**f) Find the profit function in terms of x.**

* **Profit (P) = Revenue (R) - Cost (C)**
* P(x) = R(x) - C(x)
* P(x) = (50x - 0.01x^2) - (2,500 + 4x + 0.01x^2)
* P(x) = 50x - 0.01x^2 - 2,500 - 4x - 0.01x^2
* P(x) = 46x - 0.02x^2 - 2,500

**g) Find the marginal profit.**

* **Marginal Profit:** The derivative of the profit function with respect to x.
* P(x) = 46x - 0.02x^2 - 2,500
* P'(x) = 46 - 0.04x

**h) Find P′(1,000) and P′(1,500) and interpret these quantities.**

* P'(1,000) = 46 - 0.04 * 1,000 = 46 - 40 = 6
    * When producing 1,000 bottles, the profit is increasing at a rate of $6 per additional bottle.

* P'(1,500) = 46 - 0.04 * 1,500 = 46 - 60 = -14
    * When producing 1,500 bottles, the profit is decreasing at a rate of $14 per additional bottle.
**a) Express the price p as a function of the demand x, and find the domain of this function.**

* **Solve the price-demand equation for p:**
    * x = 5,000 - 100p
    * 100p = 5,000 - x
    * p = (5,000 - x) / 100
    * p = 50 - 0.01x

* **Domain of the price function:**
    * The demand (x) must be non-negative (you cannot sell a negative number of bottles).
    * 0 ≤ x ≤ 5,000 (The maximum demand occurs when the price is 0)

**b) Find the marginal cost.**

* **Marginal Cost (MC):** The derivative of the cost function with respect to x.
    * MC = C'(x) = d/dx (2,500 + 4x + 0.01x²)
    * MC = 4 + 0.02x

**c) Find the revenue function and state its domain.**

* **Revenue (R):** Price per unit * Number of units sold
    * R(x) = p * x 
    * R(x) = (50 - 0.01x) * x 
    * R(x) = 50x - 0.01x²

* **Domain of the revenue function:** 
    * Same as the domain of the price function: 0 ≤ x ≤ 5,000

**d) Find the marginal revenue.**

* **Marginal Revenue (MR):** The derivative of the revenue function with respect to x.
    * MR = R'(x) = d/dx (50x - 0.01x²)
    * MR = 50 - 0.02x

**e) Find R′(2,000) and R′(3,000) and interpret these quantities.**

* **R'(2,000):** 
    * R'(2,000) = 50 - 0.02 * 2,000 = 50 - 40 = 10
    * Interpretation: When 2,000 bottles are sold, the revenue is increasing at a rate of $10 per additional bottle sold.

* **R'(3,000):** 
    * R'(3,000) = 50 - 0.02 * 3,000 = 50 - 60 = -10
    * Interpretation: When 3,000 bottles are sold, the revenue is decreasing at a rate of $10 per additional bottle sold.

**f) Find the profit function in terms of x.**

* **Profit (P):** Revenue - Cost
    * P(x) = R(x) - C(x)
    * P(x) = (50x - 0.01x²) - (2,500 + 4x + 0.01x²)
    * P(x) = 46x - 0.02x² - 2,500

**g) Find the marginal profit.**

* **Marginal Profit (MP):** The derivative of the profit function with respect to x.
    * MP = P'(x) = d/dx (46x - 0.02x² - 2,500)
    * MP = 46 - 0.04x

**h) Find P′(1,000) and P′(1,500) and interpret these quantities.**

* **P'(1,000):** 
    * P'(1,000) = 46 - 0.04 * 1,000 = 46 - 40 = 6
    * Interpretation: When 1,000 bottles are sold, the profit is increasing at a rate of $6 per additional bottle sold.

* **P'(1,500):** 
    * P'(1,500) = 46 - 0.04 * 1,500 = 46 - 60 = -14
    * Interpretation: When 1,500 bottles are sold, the profit is decreasing at a rate of $14 per additional bottle sold.