Question 39412
This is a workout for my calculator!
It's asking for Profit, so I rewrite
Revenue = Cost + Profit
Profit = Revenue - Cost (P = R - C)
{{{R = x(5 - .05x)}}}
{{{R = 5x - (5/100)x^2}}}
The cost of 1 box is 1.50, so the cost of x boxes is 1.5x
{{{P = R - C}}}
{{{P = 5x - (5/100)x^2 - 1.5x}}}
{{{P = -(1/20)x^2 + 3.5x}}}
If I find the roots, the maximum will be midway between 
the 2 roots. Set P = 0 to find roots.
{{{-(1/20)x^2 + 3.5x = 0}}}
multiply both sides by -1
{{{(1/20)x^2 -  3.5x = 0}}}
{{{x^2 - 70x = 0}}}
{{{x(x - 70) = 0}}}
The roots are 0,70. The midpoint is (0 + 70)/2 = 35
I want to test the equation for P to see if 35 is really
a maximum. I will find P(34), P(35), and P(36).
P(34) and P(36) should both be slightly less than P(35)
{{{P = -(1/20)x^2 + 3.5x}}}
{{{P(34) = -(1/20)34^2 + 3.5*34}}}
{{{P(34) = -57.8 + 119}}}
{{{P(34) = 61.2}}}
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{{{P(36) = -(1/20)36^2 + 3.5*36}}}
{{{P(36) = -64.8 + 126}}}
{{{P(36) = 61.2}}}
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{{{P(35) = -(1/20)35^2 + 3.5*35}}}
{{{P(35) = -61.25 + 122.5 = 61.25}}}
So, 35 boxes must be sold each month to maximize Profit,
which is $61.25