Question
What is the amount of initial investment (in thousands of dollars) needed to achieve the maximum return on investment for a startup company, given the profit function P = -2x^2 + 500x + 2,000, where P represents the profit and x represents the initial investment?
Answers
GPT-4o mini
To find the initial investment \( x \) that maximizes the profit \( P \), we need to analyze the profit function given by:
\[
P = -2x^2 + 500x + 2000
\]
This is a quadratic function in the standard form \( P(x) = ax^2 + bx + c \) where:
- \( a = -2 \)
- \( b = 500 \)
- \( c = 2000 \)
Since \( a < 0 \), the parabola opens downwards, which means there is a maximum point (vertex) that can be found using the formula for the \( x \)-coordinate of the vertex:
\[
x = -\frac{b}{2a}
\]
Substituting the values of \( a \) and \( b \):
\[
x = -\frac{500}{2 \cdot -2} = -\frac{500}{-4} = \frac{500}{4} = 125
\]
Thus, the amount of initial investment needed to achieve the maximum return on investment is \( x = 125 \) thousand dollars.
So, the final answer is:
\[
\boxed{125}
\]
\[
P = -2x^2 + 500x + 2000
\]
This is a quadratic function in the standard form \( P(x) = ax^2 + bx + c \) where:
- \( a = -2 \)
- \( b = 500 \)
- \( c = 2000 \)
Since \( a < 0 \), the parabola opens downwards, which means there is a maximum point (vertex) that can be found using the formula for the \( x \)-coordinate of the vertex:
\[
x = -\frac{b}{2a}
\]
Substituting the values of \( a \) and \( b \):
\[
x = -\frac{500}{2 \cdot -2} = -\frac{500}{-4} = \frac{500}{4} = 125
\]
Thus, the amount of initial investment needed to achieve the maximum return on investment is \( x = 125 \) thousand dollars.
So, the final answer is:
\[
\boxed{125}
\]