Question

The profit that the vendor makes per day by selling x pretzels is given by the function Upper P (x )equals negative 4 x squared plus 2 comma 400 x minus 400. Find the number of pretzels that must be sold to maximize profit.

Answer 1

The number of pretzels that must be sold to maximize profit is 300.

Explanation:

The daily profit of selling x pretzels is given by the following equation:

`P(x) = -4x^(2) + 2400x - 400`

Vertex of a quadratic function:

Suppose we have a quadratic function in the following format:

`f(x) = ax^(2) + bx + c`

It's vertex is the point
`(x_(v), f(x_(v))`

In which

`x_(v) = -(b)/(2a)`

If a<0, the vertex is a maximum point, that is, the maximum value happens at
`x_(v)`, and it's value is
`f(x_(v)`

Find the number of pretzels that must be sold to maximize profit.

This is the x of the vertex.

We have that:

`P(x) = -4x^(2) + 2400x - 400`

So
`a = -4, b = 2400`

Then

`x_(v) = -(2400)/(2*(-4)) = 300`

The number of pretzels that must be sold to maximize profit is 300.