Respuesta :
Answer:
a. R(Q) = 2OQ - 0.01Q^2
b. C(Q) = 4Q + 1000
c. MR(Q) = 20 - 0.02Q; and MC(Q) = 4.
d. The profit-maximizing quantity of books for Professor Bong to sell is 800.
Explanation:
a. What is the total revenue function R(Q) for Professor Bong's book?
From the demand function, we can obtain price function P as follows:
Q = 2,000-100P
Q + 100P = 2,000
100P = 2,000 - Q
Divide through by 100, we have:
P = 20 - (1 / 100)Q
P = 20 - 0.01Q
The total revenue function R(Q) is the product of quantity demanded Q and price function of the book P as follows:
R(Q) = Q * P = QP .................... (1)
Since P = 20 - 0.01Q, we substitute into equation (1) and solve further as follows:
R(Q) = Q(20 - 0.01Q)
R(Q) = 2OQ - 0.01Q^2 <====== Total revenue function R(Q)
b. What is the total cost function C(Q) for producing Professor Bong's book?
By removing the dollar sign, this can be obtained as follows:
MC = Marginal cost = 4
VC = Variable cost = MC * Q = 4 * Q = 4Q
FC = Fixed cost = Setup cost = 1000
Therefore, the total cost function C(Q) can be obtained as follows:
C(Q) = VC + FC ...................... (2)
Substituting the values into equation (2), we have:
C(Q) = 4Q + 1000 <====== Total cost function C(Q)
c. What are the marginal revenue function MR(Q) and the marginal cost function MC(Q)?
To obtain the marginal revenue function MR(Q), the first derivative of the total revenue function R(Q) in part a is taken as follows:
MR(Q) = R'(Q)
MR(Q) = 20 - 0.02Q
To obtain the marginal cost function MC(Q), the first derivative of the total cost function C(Q) in part b is taken as follows:
MC(Q) = C'(Q)
MC(Q) = 4
d. Find the profit-maximizing quantity of books for Professor Bong to sell.
Since profit is maximized where MR(Q) = MC(Q), the profit-maximizing quantity of books for Professor Bong to sell can be calculated as follows:
20 - 0.02Q = 4
20 = 4 + 0.02Q
20 - 4 = 0.02Q
16 = 0.02Q
Q = 16 / 0.02
Q = 800
Therefore, the profit-maximizing quantity of books for Professor Bong to sell is 800.
