Question 1198754
**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