let c(x)= 5x+49 be the cost to produce x units of a product, and let r(x)= -x^2+21x be the revenue.

Question

- Find the number of unit that maximizes the profit. - What is the maximum profit? - Find the number of items which serves as a break-even point. - What is the interval where the company will make profit?

Answer 1

c(x)=5x+49, r(x)=-x²+21x.

If p(x) is the profit, p=r-c=-x²+16x-49.

This can be written:

-49-(x²-16x)=-49-(x²-16x+64-64),

which is the same as:

-49-(x-8)²+64=15-(x-8)².

The maximum value of this expression is 15, when x=8.

So 15 is the maximum profit.

Breakeven is when r=c, so p=0, that is:

(x-8)²=15, x-8=±√15=3.87 approx.

Therefore, x=8±3.87=11.87 or 4.13.

But x must be a whole number, so if x=11, p(11)=6, and if x=12, p(12)=-1.

If x=4, p(4)=-1 and if x=5, p(5)=6.

So there is no exact breakeven point, but is close to x=4 and 12. Therefore the interval when a strict profit is being made (p>0), is between x=5 and 11, when the profit is between 6 and 15.