# The monthly cost of producing x bows is C(x)=8+2x. The revenue produced by selling x bows is R(X)=6x-0.3x^2. What is the number of bows you need to make and sell per month to make a profit?

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C(x)=8+2x. The revenue produced by selling x bows is R(X)=6x-0.3x^2.

P(x)=R(x)-C(x)

`P(x)=6x-.3x^2-8-2x`

`P(x)=-.3x^2+4x-8`

`P(x)=-(3/10)(x^2-(40/3)x+80/3)`

`=(-3/10)((x-20/3)^2-160/9)`

`=16/3-(3/10)(x-20/3)^2`

`16/3>(3/10)(x-20/3)^2`

`160/9>(x-20/3)^2`

Taking square root

`sqrt(160/9)>x-20/3`

`sqrt(160/9)+20/3>x`

`(12.65+20)/3>x`

`10.88>x`

`x<10.88`

If x=11 then P(x)=-0.3 which is loss .

If x=10 ,then P(x)=2 which is profit.

So I conclude** x may be less than or equal to 10.**