Question

The marketing department of a local shoe company found that approximately 550 pairs of running shoes are sold monthly when the price of each pair is $100. It was also observed that, for each $1 reduction in price, an additional 20 pairs of
running shoes are sold monthly. What price should the shoe store charge for a pair of running shoes in order to maximize revenue?

Answer 1

Answer:

The price that maximizes the revenue is $63.75

Step-by-step explanation:

Analyzing the statement one after the other.

$Initial\ Quantity = 550$  ---- When ----- $Initial\ Price = \ 100$

When there's a reduction of $1, we have:

$Quantity = 550 + 20x$ and $Price = 100 - x$

Where x represents the maximum reduction

At this point, we need to calculate the revenue (R)

$R = Quantity * Price.$

$R = (550 + 20x) * (100 - x)$

Open Brackets

$R = 55000 - 550x + 2000x - 20x^2$

$R = 55000 +1450x - 20x^2$

Differentiate both sides with respect to x

$\frac{dR}{dx} = 0 + 1450 - 40x$

$\frac{dR}{dx} = 1450 - 40x$

To maximize revenue;

$\frac{dR}{dx} = 0$

So:

$1450 - 40x = 0$

$1450 =40x$

Solve for x

$x = 1450/40$

$x = 36.25$

Recall that:

$Price = 100 - x$

Substitute $x = 36.25$

$Price = 100 - 36.25$

$Price = 63.75$

Hence:

The price that maximizes the revenue is $63.75