If C ( x ) is the cost of producing x units of a commodity, then the average cos
ID: 2886799 • 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) = 3,000 + 140x + 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.
dollars per unit
(c) Find the marginal cost at a production level of 1000 units.
dollars per unit
(d) Find the production level that will minimize the average cost.
units
(e) What is the minimum average cost?
dollars per unit
Explanation / Answer
We have given C(x) = 3,000 + 140x + 6x3/2,where C(x) is the cost of producing x units of a commodity.
a) We have given x=1000 units
plug x=1000 into C(x)
C(1000) = 3,000 + 140*1000 + 6(1000)3/2
=3000+140000+6(1000)3/2
=332736.65961
C(1000) = $332736.66
the total cost at a production level of 1000 units is $332,736.66
b) the average cost per unit is a(x) = C(x)/x
a(x)=C(x)/x=[3,000 + 140x + 6x3/2]/x
plug x=1000 into a(x)
a(1000)=[3,000 + 140*1000 + 6(1000)3/2]/1000
=332.73665961
a(1000)=332.73665961
the average cost at a production level of 1000 units is 332.74 dollars per unit
c) the marginal cost is C'(x)
We have C(x) = 3,000 + 140x + 6x3/2
C'(x)=0+140 + 6*(3/2)x3/2-1=140 + 9x1/2
C'(x)=140 + 9x1/2
plug x=1000 into C'(x)
C'(1000)=140 + 9(1000)1/2=424.604989415
C'(1000)=424.604989415
the marginal cost at a production level of 1000 units is 424.61 dollars per unit
d) the average cost per unit is a(x) = C(x)/x
a(x)=C(x)/x=[3,000 + 140x + 6x3/2]/x
a(x)=3000/x+140+6x3/2-1=3000/x+140+6x1/2
a(x)=3000/x+140+6x1/2
a'(x)=-3000/x^2+0+6*(1/2)*(x)-1/2
a'(x)=-3000/x^2+3/(x)1/2
to minimize the average cost we set a'(x)=0
-3000/x^2+3/(x)1/2=0
3/(x)1/2=3000/x^2
by cross multiplication
x^2/(x)1/2=3000/3=1000
x^(2-1/2)=1000
x^(3/2)=1000
x=(1000)^(1/(3/2))=(1000)^(2/3)=(10^3)^(2/3)=(10^2)=100
x=100
the production level that will minimize the average cost is 100 units
e) We have average cost a(x)=3000/x+140+6x1/2
plug x=100 units into a(x) to get the minimum average cost
a(100)=3000/100+140+6(100)1/2
=30+140+60
=230
a(100)=230
the minimum average cost is 230 dollars per unit