Answer:
a) $342,491
$342.491
$389.74
b) $400
c) $320
Step-by-step explanation:
the cost function = C(x)
C(x) = 16000 + 200x + 4x^3/2
a) when we have a unit of 1000 unit, x= 1000
C(1000) = 16000 + 200(1000) + 4(1000)^3/2
= 16000 + 200000 + 126491
= 342,491
Cost = $342,491
Average cost= C(1000) / 1000
= 342,491/1000
= 342.491
The average cost = $342.491
Marginal cost = derivative of the cost
C'(x) = 200 + 4(3/2) x^1/2
= 200 + 6x^1/2
C'(1000) = 200 + 6(1000)^1/2
= 389.74
Marginal cost = $389.74
Marginal cost = Marginal revenue
C'(x) = C(x) / x
200 + 6x^1/2 = (16000 + 200x + 4x^3/2) / x
200 + 6x^1*2 = 16000/x + 200 +4x^1/2
Collect like terms
6x^1*2 - 4x^1/2 = 16000/x + 200 -200
2x^1/2 = 16000/x
2x^3/2 = 16000
x^3/2 = 16000/2
x^3/2 = 8000
x = 8000^2/3
x = 400
Therefore, the production level that will minimize the average cost is the critical value = $400
C'(x) = C(x) / x
C'(400) = 16000/400 + 200 + 4(400)^1/2
= 40 + 200 + 80
= 320
The minimum average cost = $320
