SOLUTION: 9) The demand equation for a manufacturing product is p=20-0.25q, where q is the number of units and p is the price per unit. (a) Write the Total Revenue (TR), Average Revenue (

Question 1198754: 9) The demand equation for a manufacturing product is p=20-0.25q, where q is the number of units and p is the price per unit.
(a) Write the Total Revenue (TR), Average Revenue (AR) and Marginal Revenue (MR) equations.
(b) At what value of q will there be the total revenue at its maximum?
(c) What price should the firm charge to achieve this maximum total revenue?
(d) What is the maximum total revenue?

Answer by CPhill(1959) (Show Source): You can put this solution on YOUR website!
**a) Revenue Equations**
* **Total Revenue (TR):**
* TR = Price (p) * Quantity (q)
* TR = (20 - 0.25q) * q
* TR = 20q - 0.25q²
* **Average Revenue (AR):**
* AR = Total Revenue (TR) / Quantity (q)
* AR = (20q - 0.25q²) / q
* AR = 20 - 0.25q
* **Marginal Revenue (MR):**
* MR is the derivative of Total Revenue with respect to quantity (q):
* MR = d(TR)/dq = d(20q - 0.25q²)/dq
* MR = 20 - 0.5q
**b) Maximum Total Revenue**
* Total Revenue is maximized where the derivative of the TR function (MR) is equal to zero.
* 20 - 0.5q = 0
* 0.5q = 20
* q = 40
* **To confirm this is a maximum:**
* The second derivative of the TR function is -0.5, which is negative. This indicates that the point where MR = 0 is indeed a maximum.
**c) Price to Achieve Maximum Total Revenue**
* Substitute the value of q (40) into the demand equation:
* p = 20 - 0.25 * 40
* p = 20 - 10
* p = $10
**d) Maximum Total Revenue**
* Substitute the value of q (40) into the Total Revenue equation:
* TR = 20q - 0.25q²
* TR = 20 * 40 - 0.25 * 40²
* TR = 800 - 400
* TR = $400
**In Summary:**
* **Total Revenue (TR):** TR = 20q - 0.25q²
* **Average Revenue (AR):** AR = 20 - 0.25q
* **Marginal Revenue (MR):** MR = 20 - 0.5q
* **Quantity for Maximum Total Revenue:** q = 40 units
* **Price for Maximum Total Revenue:** p = $10
* **Maximum Total Revenue:** $400

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