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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, 23, x, squared, plus, 1258, x, minus, 8463
y=−23x
2
+1258x−8463
Answer
Attempt 1 out of 5
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, 23, x, squared, plus, 1258, x, minus, 8463
y=−23x
2
+1258x−8463
Answer
Attempt 1 out of 5
Answers
GPT-4o mini
To find the maximum amount of profit represented by the equation:
\[
y = -23x^2 + 1258x - 8463
\]
we can use the fact that this is a quadratic equation in the standard form \( y = ax^2 + bx + c \) where \( a = -23 \), \( b = 1258 \), and \( c = -8463 \).
The maximum or minimum of a quadratic function can be found at the vertex, and the x-coordinate of the vertex can be found using the formula:
\[
x = -\frac{b}{2a}
\]
Substituting the values of \( a \) and \( b \):
\[
x = -\frac{1258}{2 \times -23} = \frac{1258}{46} \approx 27.34782609
\]
Next, we calculate the maximum profit by substituting \( x \) back into the profit equation.
So we calculate:
\[
y = -23(27.34782609)^2 + 1258(27.34782609) - 8463
\]
Calculating \( 27.34782609^2 \):
\[
27.34782609^2 \approx 746.065
\]
Now calculate:
\[
y = -23(746.065) + 1258(27.34782609) - 8463
\]
Calculating each term:
1. \( -23 \times 746.065 \approx -17159.495 \)
2. \( 1258 \times 27.34782609 \approx 34328.084 \)
Now substituting these values back:
\[
y \approx -17159.495 + 34328.084 - 8463
\]
Calculating \( y \):
\[
y \approx -17159.495 + 34328.084 - 8463 \approx 1705.589
\]
Rounding this to the nearest dollar gives:
\[
\text{Maximum Profit} \approx 1706
\]
Thus, the maximum amount of profit the company can make is **$1706**.
\[
y = -23x^2 + 1258x - 8463
\]
we can use the fact that this is a quadratic equation in the standard form \( y = ax^2 + bx + c \) where \( a = -23 \), \( b = 1258 \), and \( c = -8463 \).
The maximum or minimum of a quadratic function can be found at the vertex, and the x-coordinate of the vertex can be found using the formula:
\[
x = -\frac{b}{2a}
\]
Substituting the values of \( a \) and \( b \):
\[
x = -\frac{1258}{2 \times -23} = \frac{1258}{46} \approx 27.34782609
\]
Next, we calculate the maximum profit by substituting \( x \) back into the profit equation.
So we calculate:
\[
y = -23(27.34782609)^2 + 1258(27.34782609) - 8463
\]
Calculating \( 27.34782609^2 \):
\[
27.34782609^2 \approx 746.065
\]
Now calculate:
\[
y = -23(746.065) + 1258(27.34782609) - 8463
\]
Calculating each term:
1. \( -23 \times 746.065 \approx -17159.495 \)
2. \( 1258 \times 27.34782609 \approx 34328.084 \)
Now substituting these values back:
\[
y \approx -17159.495 + 34328.084 - 8463
\]
Calculating \( y \):
\[
y \approx -17159.495 + 34328.084 - 8463 \approx 1705.589
\]
Rounding this to the nearest dollar gives:
\[
\text{Maximum Profit} \approx 1706
\]
Thus, the maximum amount of profit the company can make is **$1706**.