Problem 1. A paint manufacturer determines that the total cost in dollars of pro

ID: 1224555 • Letter: P

Question

Problem 1. A paint manufacturer determines that the total cost in dollars of producing x gallons of paint per day is C(x)=5000+x+0.001x^2

a) Find marginal cost when the production level is 500 gal per day.

b) Use the marginal cost of producing the the 501st gallon

c) Find the exact cost of producing the 501st gallon. Compare the results of b) and c).

d) Assume that the gallon of paint is sold for $5.00; find the marginal revenue and marginal profit functions.

e) Using the results of part d), find marginal revenue when the production level is 500 gal per day.

f) Using the results of part d, find marginal profit when the production level is 500 gal per day.

a) MC is the first derivative of TC with respect to quantity.

It implies MC = 1 + 0.001(2x) = 1 + 0.002x

At 500 gallons of paint per day, MC = 1 + 0.002(500) = 1 + 1 = $ 2

b) MC of producing 501 = 1 + 0.002(501) = 1 + 1.002 = $ 2.002

c) Total cost of producing 500 gallons = 5000 + 500 + 0.001(500)2 = 5750

Total cost of producing 501 gallons = 5000 + 5011 + 0.001(501)2 = 5752

So, exact cost of producing the 501st gallon = 5752 - 5750 = $ 2

d) TR = PX Q = 5x

MR = $ 5

Total Profit = TR - TC = 5x - (5000 + x + 0.001x2)

TP = 5x - 5000 - x - 0.001x2

Marginal profit = 5 - 1 - 0.002x = 4 - 0.002x

e) Marginal revenue when production level is 500 gal per day = $ 5

f) Marginal profit when production level is 500 gal per day = 4 - 0.002(500) = 4 - 1 = $ 3

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