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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, x, squared, plus, 81, x, minus, 539
y=−x
2
+81x−539
Answer
Attempt 1 out of 5
Submi
Watch Video
Show Examples
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, x, squared, plus, 81, x, minus, 539
y=−x
2
+81x−539
Answer
Attempt 1 out of 5
Submi
Answers
GPT-4o mini
To find the maximum amount of profit \( y \) from the equation
\[
y = -x^2 + 81x - 539,
\]
we can use the method of completing the square or we can find the vertex of the parabola represented by the quadratic equation.
The general form of a quadratic equation is
\[
y = ax^2 + bx + c,
\]
where \( a \) is negative (\( a = -1 \)), indicating that the parabola opens downward. The maximum value occurs at the vertex of the parabola.
The \( x \)-coordinate of the vertex can be calculated using the formula
\[
x = -\frac{b}{2a}.
\]
In this case, \( a = -1 \) and \( b = 81 \):
\[
x = -\frac{81}{2 \times -1} = \frac{81}{2} = 40.5.
\]
Now, we will substitute \( x = 40.5 \) back into the profit equation to find the maximum profit \( y \):
\[
y = - (40.5)^2 + 81(40.5) - 539.
\]
Calculating \( (40.5)^2 \):
\[
(40.5)^2 = 1640.25.
\]
Now substituting back:
\[
y = -1640.25 + 81 \times 40.5 - 539.
\]
Calculating \( 81 \times 40.5 \):
\[
81 \times 40.5 = 3285.
\]
Now substitute this value back into the equation:
\[
y = -1640.25 + 3285 - 539.
\]
Now calculate \( 3285 - 1640.25 - 539 \):
\[
3285 - 1640.25 = 1644.75.
\]
\[
1644.75 - 539 = 1105.75.
\]
Thus, the maximum profit \( y \) is approximately \( 1105.75 \). Rounding to the nearest dollar, the maximum amount of profit the company can make is
\[
\boxed{1106}.
\]
\[
y = -x^2 + 81x - 539,
\]
we can use the method of completing the square or we can find the vertex of the parabola represented by the quadratic equation.
The general form of a quadratic equation is
\[
y = ax^2 + bx + c,
\]
where \( a \) is negative (\( a = -1 \)), indicating that the parabola opens downward. The maximum value occurs at the vertex of the parabola.
The \( x \)-coordinate of the vertex can be calculated using the formula
\[
x = -\frac{b}{2a}.
\]
In this case, \( a = -1 \) and \( b = 81 \):
\[
x = -\frac{81}{2 \times -1} = \frac{81}{2} = 40.5.
\]
Now, we will substitute \( x = 40.5 \) back into the profit equation to find the maximum profit \( y \):
\[
y = - (40.5)^2 + 81(40.5) - 539.
\]
Calculating \( (40.5)^2 \):
\[
(40.5)^2 = 1640.25.
\]
Now substituting back:
\[
y = -1640.25 + 81 \times 40.5 - 539.
\]
Calculating \( 81 \times 40.5 \):
\[
81 \times 40.5 = 3285.
\]
Now substitute this value back into the equation:
\[
y = -1640.25 + 3285 - 539.
\]
Now calculate \( 3285 - 1640.25 - 539 \):
\[
3285 - 1640.25 = 1644.75.
\]
\[
1644.75 - 539 = 1105.75.
\]
Thus, the maximum profit \( y \) is approximately \( 1105.75 \). Rounding to the nearest dollar, the maximum amount of profit the company can make is
\[
\boxed{1106}.
\]