Applications of Integration

Question

A company estimates that the marginal cost (in dollars per item) of producing x items is dx/dy= 1.8-0.06x. If the cost of producing one item is $575, find the cost function C(x), then find the cost of producing 100 items.

Find the cost function C(x)=____

Find the cost of producing 100 items= $____

Answer 1

Marginal Cost = MC = 1.8-0.06x

Total Cost = the integral of MC = 1.8x -0.03x^2 + k

where k is an unknown constant

C(1) = 1.8(1) -0.03(1)^2 +k

575 = 1.8-.03 + k

k = 575-1.8+.03 = 573.23

k = 573.23= Fixed Cost

C(x) = 1.8x -0.03x^2 + 573.23

C(100) = 1.8(100) -0.03(100)^2 +573.23

= 180 - 300 +573.23

=$453.23

for cost of 100 items

the squared term reflects some economies of scale, but this Cost function has to be limited to a range of production levels, as it can't be true for much higher production levels, At some point this would mean

there is no cost to produce large volumes of items.

Or this "cost" function might really be a profit function, where you incur iniitial losses.

Usually these problems are little more realistic. The numbers are strange enough that maybe there's a typo or problem in the problem. Marginal Cost does = dy/dx not dx/dy

I assumed you meant dy/dx