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 : $C(x) =\frac{5}{2}x^2+120 x+560$  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 :

$C(x) =\frac{5}{2}x^2+120 x+560$  

Now, Marginal cost = $\frac{d}{dx}C(x)$

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 : $\frac{d}{dx}C(x)= \ 210$

$\frac{d}{dx}C(x)= \frac{5}{2}(2x)+120 = 210$

$5x + 120 = 210$

$5x=210-120$

$5x=90$

$x=45$

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