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The weekly cost of manufacturing x telephones per week is found by a manufacturer to be C(x) = 500 + 20x = x^2 dollars. the telephone can be sold at a price p = \$80 each. find the manufacturing break-even production level and for what production levels will the manufacturer experience a profit

telephones are not cheap to make, young whoppa

breakeven occurs when cost = revenue
don't know whether you have + or minus x^2. On my keyboard, + and = are the same key, so I'll assume "+".

x^2 + 20x + 500 = 80x
x^2 - 60x + 500 = 0
(x-10)(x-50) = 0

So, for x between 10 and 50, cost is less than revenue.

This is a strange model. Usually as quantity increases, cost goes down.

Maybe you should have had C(x) = 500 + 20x - 1/x^2
or something. That would mean there's a fixed cost of \$500 just for making phones, and a \$20/phone cost for materials, say, and a decreasing manufacturing cost as quantity goes up. (efficiency of scale)

In that case, we'd have

500 + 20x - 1/(x^2+1) = 80x

That shows costs greater than revenue until x = 8.3 or so, then revenue is greater than costs.

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