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

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Question 1199380: 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. (2 marks)
(b) At what value of q will there be the total revenue at its maximum? (4 marks)
(c) What price should the firm charge to achieve this maximum total revenue? (2 marks)
(d) What is the maximum total revenue? (2 marks)

Answer by ikleyn(52781)   (Show Source): You can put this solution on YOUR website!
.
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. (2 marks)
(b) At what value of q will there be the total revenue at its maximum? (4 marks)
(c) What price should the firm charge to achieve this maximum total revenue? (2 marks)
(d) What is the maximum total revenue? (2 marks)
~~~~~~~~~~~~~~~~~~

(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).

Solved. All questions are answered, with full explanations.



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