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19. The cost function of a product is C(Q) = 3Q² +8 and the demand function of the same product is P =1/3 Q² — 10Q + 105, where Q is output, P the price per unit output. a) Find the marginal cost when output is 4 b) Find the marginal revenue at the output level 4. c) Find the marginal profit at the output level 4. d) Also find the maximum profit.
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Respuesta :

The marginal revenue at the output level 4 is 24, marginal revenue at the output level 4 is 41, and marginal profit at the output level 4 is 17.

What is a marginal cost?

It is defined as the cost showing an increase in the cost when the number of units produced increases, In simple words it is the ratio of the cost to quantity.

We have a cost function of a product:

C(Q) = 3Q² +8

a) To find the marginal cost to differentiate it with respect to Q and plug

Q = 4:

C'(Q) = 6Q

C'(4) = 6(4) = 24

b) R(Q) = P×Q

[tex]\rm R(Q) = (\dfrac{1}{3} Q^2 - 10Q + 105)Q[/tex]

[tex]\rm R(Q) = \dfrac{1}{3} Q^3 - 10Q^2 + 105Q[/tex]

R'(Q) = Q² - 20Q + 105

Plug Q = 4

R'(Q) = (4)² - 20(4) + 105

R'(Q) = 41

c) Marginal profit:

MP(Q)  = R(Q) - C(Q)

After calculating:

[tex]\rm MP(Q) = \dfrac{1}{3}Q^3-13Q^2+105Q-8[/tex]

MP'(Q) = Q² - 26Q + 105

Plug Q = 4

MP'(Q) = 16 - 104 + 105 = 17

Similar, we can find the maximum profit.

Thus, the marginal revenue at the output level 4 is 24, marginal revenue at the output level 4 is 41, and marginal profit at the output level 4 is 17.

Learn more about the marginal cost here:

brainly.com/question/7781429

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