Question

A model is used to predict monthly revenue for a new sports drink. In the model, x is the number of $1-price increases from the original $2 per bottle price.

Model A: f(x)=-12.5x^2+75x+200

Using Model A, the price should be $___ to maximize monthly revenue because vertex/y-intercept/x-intercept occurs at x=___

Answer 1

To find the price that maximizes the monthly revenue using Model A, we first need to determine the vertex of the quadratic function given by the formula:

\[
f(x) = -12.5x^2 + 75x + 200.
\]

In a quadratic equation of the form \(ax^2 + bx + c\), the x-coordinate of the vertex is given by the formula:

\[
x = -\frac{b}{2a}.
\]

In this case:

- \(a = -12.5\)
- \(b = 75\)

Plugging in the values:

\[
x = -\frac{75}{2 \times -12.5} = -\frac{75}{-25} = 3.
\]

So, the number of $1-price increases from the original $2 price is \(x = 3\).

Next, we calculate the new price. The original price is $2, and since \(x\) is the number of $1-price increases, the price will be:

\[
\text{New Price} = 2 + x = 2 + 3 = 5.
\]

To summarize:

- The price should be **$5** to maximize monthly revenue.
- The vertex occurs at \(x = 3\).

Thus, the complete answer is:

The price should be **$5** to maximize monthly revenue because the vertex occurs at \(x = 3\).