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

Answer:

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

Step-by-step 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

$x_{v} = -\frac{b}{2a}$

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

$x_{v} = -\frac{2400}{2*(-4)} = 300$

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