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:

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, [tex]p[/tex] is the price of a cookie and the number of cookies sold is [tex]x =- 2000 p + 4000[/tex].

[tex]\text{Revenue}=p(-2000 p + 4000)[/tex]

[tex]\text{Revenue}=-2000 p^2 + 4000p[/tex]

The above equation is a quadratic equation and the graph of the above equation will be a downward parabola because the coefficient of [tex]p[/tex] is negative.

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

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

By compare Revenue equation with [tex]ax^2+bx+c=0[/tex], we get:

[tex]x=p, a=-2000, b=4000 \text{ and } c=0[/tex]

Now put the respective values in the formula [tex]x=\frac{-b}{2a}[/tex] .

[tex]p=\frac{-4000}{2(-2000)}[/tex]

[tex]p=1[/tex]

So revenue will be maximum for [tex]p=1[/tex].

Now put [tex]p=1[/tex] in [tex]\text{Revenue}=-2000 p^2 + 4000p[/tex]

[tex]\text{Revenue}=-2000 (1)^2 + 4000(1)[/tex]

[tex]\text{Revenue}=-2000 + 4000[/tex]

[tex]\text{Revenue}=2000[/tex]

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

Maximum revenue generated = $2000

Price at which the maximum revenue is generated = $1

Steps to find the maximum revenue,

Given in the question,

        x = -2000p + 4000

Expression for the revenue generated = Price of one cookie × Number of cookies sold

R = p(-2000p + 4000)

R = -2000p² + 4000p

For the maximum revenue,

"Find the derivative of the expression for revenue and equate it to zero"

R' = -4000p + 4000 = 0

4000p = 4000

p = $1

For p = 1,

R = -2000(1) + 4000

R = $2000

       Therefore, for the price of one cookie as $1, maximum revenue generated will be $2000.

Learn more about the maximization here,

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