Question

Luci Lulu opened a cookie store in the mall. She found that the relationship between the price of a cookie, p, and the number of cookies sold, x, is given by the linear relationship x = −2000 p + 4000 . Find the maximum revenue Luci can make selling cookies in one day. Find the price she should sell the cookies for to make the maximum revenue.

Answer 1

Answer:

The price to make maximum revenue is 1 and the maximum revenue is 2000.

Step-by-step explanation:

Consider the provided information.

Revenue is the product of the number of sales and the sales price.

Here, $p$ is the price of a cookie and the number of cookies sold is $x =- 2000 p + 4000$.

$\text{Revenue}=p(-2000 p + 4000)$

$\text{Revenue}=-2000 p^2 + 4000p$

The above equation is a quadratic equation and the graph of the above equation will be a downward parabola because the coefficient of $p$ is negative.

The vertex(axis of symmetry) of a downward parabola has the maximum point.

The axis of symmetry is  $x=\frac{-b}{2a}$  for the quadratic equation $ax^2+bx+c=0$.

By compare Revenue equation with $ax^2+bx+c=0$, we get:

$x=p, a=-2000, b=4000 \text{ and } c=0$

Now put the respective values in the formula $x=\frac{-b}{2a}$ .

$p=\frac{-4000}{2(-2000)}$

$p=1$

So revenue will be maximum for $p=1$.

Now put $p=1$ in $\text{Revenue}=-2000 p^2 + 4000p$

$\text{Revenue}=-2000 (1)^2 + 4000(1)$

$\text{Revenue}=-2000 + 4000$

$\text{Revenue}=2000$

Hence, the price to make maximum revenue is 1 and the maximum revenue is 2000.