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I have no idea how to do this, I tried a lot of things but they don't make sense and I have too many variables.

A manufacturer has been selling lamps at the price of \$6/lamp, and at this price they have been selling 3000 lamps a month. The manufacturer wishes to raise the price and estimated that for each \$1 increase they will sell 1000 fewer lamps a month. The manufacturer can produce the lamps at a cost of \4 per lamp Express the manufacturers monthly profit as a function of the price that the lamps are sold, draw the graph and estimate the optimal selling point.

I think the profit should be \#(\mathrm{lamps\ sold})\cdot(\mathrm{price\ of\ lamps}) - 4\cdot\#(\mathrm{lamps\ sold})$.

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If the manufacturer decides to set her price to \$p$ then her estimation of the number of sold lamps per month is $ 3000-(p-6)1000=9000-1000p. $ Hence the monthly profit is $ (9000-1000p)(p-4)=-1000p^2+13000p-36000. $ Computing the derivative and setting it to zero shows that the optimal selling point is p^*=6.5$.

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    I give up, I just can't do word problems and I am wasting my time trying to study these. There are other parts of the test I will fail if I don't study that, thanks though.2011-10-22