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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 3 out of 5

Answer 1

To find the maximum amount of profit \( y \) that the company can make based on the given quadratic profit equation:

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

we can use the method of finding the vertex of a parabola. The vertex form of a quadratic equation is represented as \( y = ax^2 + bx + c \), where the maximum or minimum occurs at:

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

In this equation:
- \( a = -33 \)
- \( b = 1612 \)

Substituting the values into the vertex formula:

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

Now, we need to plug this value of \( x \) back into the profit equation to find the maximum profit \( y \):

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

Calculating \( (24.4)^2 \):

\[
(24.4)^2 \approx 595.36
\]

Then substituting:

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

Calculating each term:

\[
-33(595.36) \approx -19666.88
\]
\[
1612 \times 24.4 \approx 39308.8
\]

Now summing these values:

\[
y \approx -19666.88 + 39308.8 - 11738
\]
\[
y \approx -19666.88 + 39308.8 \approx 19641.92
\]
\[
y \approx 19641.92 - 11738 \approx 7913.92
\]

Rounding to the nearest dollar, the maximum profit the company can make is:

\[
\boxed{7914}
\]