Question

approximates the dollar cost of producing x units of a product. The manu- facturer believes it cannot make a profit when the marginal cost goes beyond $210. What is the most units the manufacturer can produce and still make a profit? What is the total cost at this level of production?

Answer 1

The question is incomplete. The complete question is :

A manufacturer believes that the cost function :
`$C(x) =(5)/(2)x^2+120 x+560$` approximates the dollar cost of producing x units of a product. The manu- facturer believes it cannot make a profit when the marginal cost goes beyond $210. What is the most units the manufacturer can produce and still make a profit? What is the total cost at this level of production?

Solution :

Given the cost function is :

`$C(x) =(5)/(2)x^2+120 x+560$`

Now, Marginal cost =
`$(d)/(dx)C(x)$`

So, if the marginal cost = $ 210, then the manufacturer also makes a profit and if it goes beyond $ 210 than the manufacturer cannot make a profit.

Therefore, we have to equate :
`$(d)/(dx)C(x)= \$ 210$`

`$(d)/(dx)C(x)= (5)/(2)(2x)+120 = 210$`

`$5x + 120 = 210$`

`$5x=210-120$`

`$5x=90$`

`$x=45$`

So when x = 45, then C(x) = $ 8042.5

Therefore, the manufacturer
`$\text{can make up}$` to 45 units and
`$\text{still makes a profit.}$` This leads to a total cost of $ 8042.5