a shoe retailer finds that she can sell 23 pairs per month when the price is $43 per pair and that for each dollar by which she lowers the price, her monthly sales increase by 5 pairs. If the shoes cost her $30 a pair wholesale, what is the correct expression for her monthly profits in terms of her selling price per pair?
The basic profit equation is:
P = ($43)(23 pairs) - ($30)(23 pairs) = ($13)(23 pairs)
Lowering the price by $x increases sales by 5x pairs of shoes, so the modified equation for monthly profit is:
P = ($13-$x)(23+5x) = 299 + 42x - 5x2
To find the maximum profit, take the derivative of P wrt x, set it zero, and solve for x:
dP/dx = 42 - 10x
0 = 42 - 10x
10x = 42
x = 4.2
The retailer will maximize her profit with a selling price of $43 - $4.20 = $38.80. She will sell 23 + 5(4.2) = 44 pairs of shoes. Her profit will be $387.20.
You can also solve this algebraically by noting that the modified profit equation is an inverted parabola, so the maximum point will be the vertex, which is located at the point:
x = -b/2a = (-42)/(-10) = 4.2