Question

A model for a company's revenue is R=-15p^2+300p+12,000, where p is the price in dollars of the company's product. What prize will maximize revenue?

Answer 1

Check the picture below.

$\bf \textit{ vertex of a vertical parabola, using coefficients}\\\\ \begin{array}{lccclll} R = &{{ -15}}p^2&{{ +300}}p&{{ +12,000}}\\ &\uparrow &\uparrow &\uparrow \\ &a&b&c \end{array}\qquad \left(-\cfrac{{{ b}}}{2{{ a}}}\quad ,\quad {{ c}}-\cfrac{{{ b}}^2}{4{{ a}}}\right)$

so the Revenue will be the highest at   $\bf {{ c}}-\cfrac{{{ b}}^2}{4{{ a}}}$   and that will happen at the price of    $\bf -\cfrac{{{ b}}}{2{{ a}}}$

Answer 2

In order to do this you have to complete the square to get the vertex of the equation. The vertex will tell you the max value or the min value of whatever it is we are looking for. Here, our x values are the prices of whatever we are selling and the y values are the revenue dollars. Completing the square looks like this for us:
$R(p)=-15 p^{2}+300p+12,000$
Move the 12,000 over to the other side and set the equation equal to 0:
$-15 p^{2}+300p=-12,000$
In order to complete the square, the leading coefficient on the squared term has to be a 1 and it's a -15, so we have to factor that out:
$-15( p^{2}-20p)=-12,000$
Now it's ready to complete the square on it. Do this by taking half the linear term, squaring it, and then adding it in. Our linear term is 20p.  Half of 20 is 10 and 10 squared is 100, so we will add in 100 on the left. Let's do this next step in two parts:
$-15( p^{2}-20p+100)=-12,000$
That's not quite complete yet because if we add in 100 inside the parethesis on the left we have to add in that same amount on the right. But on the left, we didn't just add in 100, we added in -15 TIMES 100 cuz we can't just forget about the -15 we factored out.  That looks like this then:
$-15( p^{2}-20p+100)=-12,000-1,500$
The whole reason for doing this is to create a perfect square binomial on the left which is what we have done. The perfect square binomial is this (and we are going to do the subtraction on the right at the same time):
$-15(p-10)^{2}=-13,500$
If we move the right side over to the left we get this:
$-15(p-10) ^{2}+13,500=0$
which gives us a vertex of (10, 13,500).  That means that at a price of $10, the max revenue will be $13,500.  See how beautifully that works out!??