1. The vertex of this function will tell us both the maximum profit and how many units are sold to yield this profit. Using \(a=-0.01\) and \(b=520\text{,}\) we have:

    \begin{align*} n\amp=-\frac{b}{2a}\\ \amp=-\frac{520}{2(-0.01)}\\ \amp=26000 \end{align*}

    Now we will find the value of \(P\) when \(n=26000\text{:}\)

    \begin{align*} P\amp=-0.01(\substitute{26000})^2+520(\substitute{26000})-54000\\ \amp=6706000 \end{align*}

    The maximum profit is \(\$6{,}706{,}000\text{,}\) which occurs if \(26{,}000\) units are sold.

  2. To find how many refrigerators need to be sold in order for the company to “break even”, we will set \(P=0\) and solve for \(n\) using the quadratic formula:

    \begin{align*} 0\amp=-0.01n^2+520n-54000\\ n\amp=\frac{-520\pm \sqrt{520^2-4(-0.01)(-54000)}}{2(-0.01)}\\ n\amp\frac{-520\pm\sqrt{268240}}{-0.02}\\ n\amp\approx 104, n\approx 51896 \end{align*}

    The company will break even if they sell about \(104\) refrigerators or \(51{,}896\) refrigerators.