Suppose the total cost (in dollars) of manufacturing x units of a certain commodity is
C(x)=3x^2+16x+300
At what level of production is the average cost per unit the smallest. ___________________.
At what level of production is the average cost per unit equal to the marginal cost. ________________
2007-11-19 17:22:16 · 3 answers · asked by Ksyha 1 in Science & Mathematics ➔ Mathematics
Average cost = C(x) / x
= (3x^2+16x+300)/x
A(x) = 3x + 16 + 300/x
Take the derivative of A(x) with respect to x and set equal to zero to find the value which minimizes A.
A'(x) = 3 - 300/x^2 = 0
300/x^2 = 3
x^2 = 100
x = 10
So the average cost per unit is smallest manufacturing 10 units.
The marginal cost is the change in cost after increasing production by one unit and is the derivative of total cost.
M(x) = C'(x) = 6x + 16
and we want to find when A(x) = C'(x)
3x + 16 + 300/x = 6x + 16
300/x = 3x
x^2 = 100
x = 10 again.
2007-11-19 17:41:04 · answer #1 · answered by Scott R 6 · 1⤊ 0⤋
Average cost= C(x)/x = 3x+16 + 300/x
Take derivitive to find minimum or 3 - 300/(x^2)=0
Solving, x=10
Marginal cost is cost of next item. If current output is x items, the marginal cost is
3(x+1)^2 + 16(x+1) - 3x^2 -16x .
This simplifes somewhat.
The average cost is given above. Equate the two relations to find the desired level of production.
2007-11-20 01:43:03 · answer #2 · answered by cattbarf 7 · 1⤊ 0⤋
the smallest would be the minimum value of the parabola... aka the y value of the vertex... to find the X value of the vertex use the formula X=-B/2A... and then plug this into the equation.... the value at which it is equal would be when x and y are equal...
Cost per unit is the least... 278.66666666
equal... would be 300 because X=0
2007-11-20 01:29:00 · answer #3 · answered by Anonymous · 0⤊ 1⤋