If C ( x ) is the cost of producing x units of a commodity, then the average cos
ID: 2872966 • Letter: I
Question
If C(x) is the cost of producing x units of a commodity, then the average cost per unit is a(x) = C(x)/x. Consider the C(x) given below. Round your answers to the nearest cent.
C(x) = 24,000 + 80x + 6x3/2
(a) Find the total cost at a production level of 1000 units.
$ 1
(b) Find the average cost at a production level of 1000 units.
2 dollars per unit
(c) Find the marginal cost at a production level of 1000 units.
3 dollars per unit
(d) Find the production level that will minimize the average cost.
4 units
(e) What is the minimum average cost?
5 dollars per unit
Explanation / Answer
C(x) = 24,000 + 80x + 6x3/2
a) x = 1000
==> C(1000) = 24,000 + 80(1000) + 6(1000)3/2
==> Cost of producing 1000 units = $ 293736.65
b) Average cost of production of 1000 units = C(1000)/1000
= 293736.65/1000
= $ 293.74
c) Marginal cost ==> C '(x)
==> M(x) = 0 + 80 + 6(3/2)x3/2 -1
==> M(x) = 80 + 9x1/2
==> M(1000) = 80 + 9(1000)1/2
==> M(1000) = 364.60
Marginal cost = $ 364.60
d) average cost = C(x)/x
for minimal cost d/dx C(x)/x = 0
==>d/dx (24,000 + 80x + 6x3/2)/x = 0
==> d/dx 24,000x-1 + 80 + 6x1/2= 0
==> -24000x-2 + 0 + 6(1/2)x-1/2 = 0
==> -24000 + 0 + 6(1/2)x3/2 = 0
==> 3x3/2 = 24000
==> x3/2 = 8000
==> x = 400
Production of 400 units minimize average cost
e) minimum average cost = (24000 +80(400) + 6(400)3/2)/400
==> $260