Question

A company sells widgets. The amount of profit, y, made by the company, is related to the selling price of each widget, x, by the given equation. Using this equation, find out the maximum amount of profit the company can make, to the nearest dollar.
y, equals, minus, 33, x, squared, plus, 1612, x, minus, 11738
y=−33x
2
+1612x−11738
Answer
Attempt 1 out of 5

Submit

Answer 1

To find the maximum profit using the given quadratic profit equation:

\[
y = -33x^2 + 1612x - 11738
\]

This is a quadratic equation in the form \(y = ax^2 + bx + c\), where \(a = -33\), \(b = 1612\), and \(c = -11738\). Since \(a < 0\), the parabola opens downward, meaning there is a maximum point at the vertex.

The x-coordinate of the vertex for a quadratic equation is given by the formula:

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

Substituting the values of \(a\) and \(b\):

\[
x = -\frac{1612}{2 \cdot -33} = \frac{1612}{66} \approx 24.4
\]

Now, we will substitute \(x \approx 24.4\) back into the profit equation to find the maximum profit \(y\).

Calculating:

\[
y = -33(24.4)^2 + 1612(24.4) - 11738
\]

Calculating \( (24.4)^2 \):

\[
(24.4)^2 = 595.36
\]

Now substituting:

\[
y \approx -33(595.36) + 1612(24.4) - 11738
\]

Calculating \( -33(595.36) \):

\[
-33(595.36) \approx -19666.88
\]

Calculating \( 1612(24.4) \):

\[
1612(24.4) \approx 39288.8
\]

Now substituting back into the equation for \(y\):

\[
y \approx -19666.88 + 39288.8 - 11738
\]

Calculating:

\[
y \approx 39288.8 - 19666.88 - 11738 \approx 7843.92
\]

Rounding to the nearest dollar:

\[
\text{Maximum Profit} \approx 7844
\]

Thus, the maximum amount of profit the company can make is **$7844**.