If C ( x ) is the cost of producing x units of a commodity, then the average cos
ID: 2864069 • Letter: I
Question
If C(x) is the cost of producing x units of a commodity, then the average cost per unit is c(x) = C(x)/x. Consider the cost function C(x) given below. (Round your answers to the nearest cent.)
C(x) = 24,000 + 210x + 6x3/2
(a) Find the total cost at a production level of 1000 units.
(b) Find the average cost at a production level of 1000 units.
(c) Find the marginal cost at a production level of 1000 units.
(d) Find the production level that will minimize the average cost.
(e) What is the minimum average cost?
Explanation / Answer
a. x = 1000 so plug x into equation C(x)
C(1000) = 24000+210*1000+6*10003/2 = 423736.66
b. the average cost per unit is c(x) = C(x)/x
c(x) = 423736.66/1000 = 423.73 / unit.
c. Marginal Cost = the derivative of Total Cost with respect to Quantity
= dTC/dQ = 210+ 3/2*(6x0.5)
So for x = 1000,
Marginal Cost = 210 + 9*(10000.5) = $494.6 for an additional unit produced.
d) Average cost c(x) = C(x) /x = 24000/x +210 + 6x1/2
to minimize c(x) , dc(x)/dx = -24000/x2 + 6*1/2 *x-1/2 =0
x = 400 units
At production of 400 units, average cost will be minimum.
e) At x=400, c(400) = 24000/400 +210 + 6*4000.5 = 390