Calculus Question

Question

The cost function for a product is C(q)=q3−75q2+1875q+1120 for 0≤q≤50 and a price per unit of $579. 

Round your answers to the nearest whole number. 

a) What production level maximizes profit? 

q = 

b) What is the total cost at this production level? 

cost = $ 

c) What is the total revenue at this production level? 

revenue = $ 

d) What is the total profit at this production level? 

profit = $ 

Answer 1

The following involves some tedious calculations. With luck maybe they're right.

IF so, the answer is produce zero, and incur only the fixed costs of $1,120

the basic method is take the derivative of the cost function, set it equal to zero and solve for q

then plug that q value into the cost function, then compare it with both C(0) and C(50), the end points.

Seems to turn out that C(0) < C(25) < C(50), same with profit levels

Price = AR = Average Revenue = per unit revenue = 579

Cost = C = q^3 -75q^2 + 1875q + 1120

Take the derivative of Cost and set it equal to zero

C'(q) = 3q^2 -150q + 1875 = 0, divide by 3

= q^2 - 50q + 625 = 0

= q^2 -50q = -625

= q^2 -50q + 625 =- 625 + 625 =0

= (q-25)^2 = 0

= q=25

C(25) = 25^3 - 75(25)^2 + 1875(25) + 1120= 16,745

C(0) = fixed cost = 1120

C(50) = (50)^3 - 75(50)^2 + 1875(50) + 1120 = 31,250

cost minimizing output level = 0

Profit(0) =-1120

Profit(25) = 579(25) - 16,745 = 14,475 - 16,745 = -2270 worse than shutting down, slightly over twice as bad

Profit(50) = 14475- 31,250 = -16775 far worse than q=25 or q=0

Profit maximizing (or really loss minimizing) output level = q=0, just shutdown

at that production level, total cost = fixed cost = $1,120, variable costs =0

total revenue = 0

total profit = negative 1120

the cost curve is cubic, which usually has a local max and min, but this curve has neither, just an inflection point at q=25. cost is everywhere increasing. Cost minimizing q = 0