Answer
The maximum profit: $88500$
$x=15000$ units must be sold to generate the maximum profit
Work Step by Step
$P(x)=-0.0004x^2+12x-1500$,
To find the maximum profit, we can turn an equation from a standard form into a vertex form by completing the square.
$P(x)=-0.0004(x^2-30000x)-1500$,
$-0.0004(x^2-30000x+225000000)-1500$,
for an equation to match the previous one just as we added
$-0.0004(225000000)$ into the equation
we need to subtract
$-0.0004(225000000)$ from an equation,
thus,$P(x)=-0.0004(x^2-30000x+225000000)-1500-(-0.0004(225000000))$
$P(x)=-0.0004(x-15000)^2+88500$,
Therefore, we can see that from an equation the maximum profit is $88500$ and $x=15000$ units must be sold to generate the maximum profit.
