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 Revenu

(a)  The total revenue is the product of the number of items by the price per item

         R(p,q) = q*p = q*(20-0.25q).    (1)


     The average revenue is the ratio of the total revenue to the number of items,
     so the average revenue is the same as the price per item

         average revenue = p.


     What is the marginal revenue, I don't know, since I never studied Finance in my life.
     From the other side, I assume that the post came from a person who studies it,
     so this person must know it much better than I do.  
             
             (This question is of the level "how much is 2x2" in Finance.)



(b)  To answer question (b), notice that the revenue (1) is a quadratic function of q, presented
     as the product of two linear binomial, q and (20-0.25q).

     Therefore, this quadratic function has the zeroes  q=0  and  q= = 80.

     This quadratic function gets the maximum precisely half-way between the zeroes, i.e.
     at  q= 40.  It is the answer to question (b).



(c)  In (b), we just found that the maximum revenue is achieved at q= 40.

     Due to the formula  p = 20 - 0.25q,  it gives  p = 20 - 0.25*40 = 20 - 10 = 10.

     It is the answer to question (c).



(d)  We just know from (b) that the maximum revenue is achieved at q= 40.

     So, we substitute q= 40 into formula (1), and we get

         the maximum revenue = 40*(20-0.25*40) = 40*(20-10) = 40*10 = 400.

     It is the answer to question (d).