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, 49, x, squared, plus, 1875, x, minus, 10599
y=−49x
2
+1875x−10599
Answer
Attempt 1 out of 5
y, equals, minus, 49, x, squared, plus, 1875, x, minus, 10599
y=−49x
2
+1875x−10599
Answer
Attempt 1 out of 5
Answers
GPT-4o mini
To find the maximum amount of profit, we need to analyze the quadratic equation given:
\[ y = -49x^2 + 1875x - 10599 \]
This is a quadratic equation in the form \( y = ax^2 + bx + c \), where:
- \( a = -49 \)
- \( b = 1875 \)
- \( c = -10599 \)
Since \( a < 0 \), the parabola opens downwards, indicating that there is a maximum point.
The x-coordinate of the vertex of a parabola given by the equation \( y = ax^2 + bx + c \) can be found using the formula:
\[ x = -\frac{b}{2a} \]
Plugging in the values of \( a \) and \( b \):
\[ x = -\frac{1875}{2 \times -49} \]
\[ x = \frac{1875}{98} \]
\[ x \approx 19.1 \]
Next, we substitute this value back into the equation to find the maximum profit \( y \):
\[ y = -49(19.1)^2 + 1875(19.1) - 10599 \]
Calculating \( (19.1)^2 \):
\[ (19.1)^2 \approx 364.81 \]
Now substituting this into the equation:
\[ y = -49(364.81) + 1875(19.1) - 10599 \]
\[ y \approx -17895.69 + 35782.5 - 10599 \]
\[ y \approx -17895.69 + 25183.5 \]
\[ y \approx 7287.81 \]
Rounding to the nearest dollar, the maximum profit the company can make is:
\[
\boxed{7288}
\]
\[ y = -49x^2 + 1875x - 10599 \]
This is a quadratic equation in the form \( y = ax^2 + bx + c \), where:
- \( a = -49 \)
- \( b = 1875 \)
- \( c = -10599 \)
Since \( a < 0 \), the parabola opens downwards, indicating that there is a maximum point.
The x-coordinate of the vertex of a parabola given by the equation \( y = ax^2 + bx + c \) can be found using the formula:
\[ x = -\frac{b}{2a} \]
Plugging in the values of \( a \) and \( b \):
\[ x = -\frac{1875}{2 \times -49} \]
\[ x = \frac{1875}{98} \]
\[ x \approx 19.1 \]
Next, we substitute this value back into the equation to find the maximum profit \( y \):
\[ y = -49(19.1)^2 + 1875(19.1) - 10599 \]
Calculating \( (19.1)^2 \):
\[ (19.1)^2 \approx 364.81 \]
Now substituting this into the equation:
\[ y = -49(364.81) + 1875(19.1) - 10599 \]
\[ y \approx -17895.69 + 35782.5 - 10599 \]
\[ y \approx -17895.69 + 25183.5 \]
\[ y \approx 7287.81 \]
Rounding to the nearest dollar, the maximum profit the company can make is:
\[
\boxed{7288}
\]