A store is demoing different prices for one of its items. When the price of the item was $10 per item, the store sold 24 items per day. For every $1 increase to the item price, they sell 1 less items each day. To maximize the revenue, how much, in dollars, should they charge for 1 item? (Revenue = price per item x number of items sold).

Summary Submit

We can make a table of values and inspect which row provides the highest revenue.

$$ \begin{array} {|c|c|c|} \hline \text{Price per item} & \text{number of items} & \text{Revenue} \\ \hline \$10 & 24 & \$10\cdot24=\$240 \\ \hline \$11 & 23 & \$11\cdot23=\$253 \\ \hline \$12 & 22 & \$12\cdot22=\$264 \\ \hline \$13 & 21 & \$13\cdot21=\$273 \\ \hline \$14 & 20 & \$14\cdot20=\$280 \\ \hline \$15 & 19 & \$15\cdot19=\$285 \\ \hline \$16 & 18 & \$16\cdot18=\$288 \\ \hline \$17 & 17 & \$17\cdot17=\colorbox{aqua}{$\$289$} \\ \hline \$18 & 16 & \$16\cdot18=\$288 \\ \hline \$19 & 15 & \$17\cdot17=\$285 \\ \hline \$20 & 14 & \$16\cdot18=\$288 \\ \hline \$21 & 13 & \$17\cdot17=\$285 \\ \hline \end{array} $$

We see that the maximum revenue occurs when the price per item is \(\$17 \) per item.

Note that there was a general pattern used to create the table above. For every $1 increase in the price of the item, the number of items increased by 1. Therefore, for every \(\$x\) we increase the price, the number of items should decrease by \(x\) as well.

$$ \text{ Revenue} = \text{ Price per item} \cdot \text{ Number of items}$$ $$ R=(10+x)(24-x) $$

This is the factored form of a quadratic equation with zeroes at \(-10\) and \(24\). Expanding the equation will result in a quadratic equation with a negative leading coefficient, which means the parabola opens downward.

$$ R=-x^2+14x+240 $$

The \(x\) value that maximizes the value of the parabola must occur at the vertex. Since the \(x\)-coordinate of the vertex is the midpoint of the zeroes,

$$ x = \frac{-10+24}{2} $$ $$ x = \frac{14}{2} $$ $$ x=7$$

Sidenote: you can also use \(-\dfrac{b}{2a}\) to obtain the \(x\)-coordinate of the vertex.

When \(x=7\), the price per item is \(10+7=\boxed{17}\)