Question
Question
The profit (in thousands of dollars) of a company is represented as P=−5x^2+1,000x+5,000
, where P
represents the profit and x represents the amount spent on marketing (in thousands of dollars). How much spending in the thousands will be directed toward marketing to achieve the maximum profit?(1 point)
$
The profit (in thousands of dollars) of a company is represented as P=−5x^2+1,000x+5,000
, where P
represents the profit and x represents the amount spent on marketing (in thousands of dollars). How much spending in the thousands will be directed toward marketing to achieve the maximum profit?(1 point)
$
Answers
GPT-4o mini
To find the amount spent on marketing that maximizes the profit \( P \), we can use the formula for the vertex of a quadratic equation. The given profit function is:
\[
P = -5x^2 + 1000x + 5000
\]
This is a quadratic equation in the form \( P = ax^2 + bx + c \), where \( a = -5 \), \( b = 1000 \), and \( c = 5000 \).
The x-coordinate of the vertex of a parabola described by the equation \( ax^2 + bx + c \) is given by the formula:
\[
x = -\frac{b}{2a}
\]
Substituting the values of \( a \) and \( b \):
\[
x = -\frac{1000}{2 \times -5} = -\frac{1000}{-10} = 100
\]
Thus, the amount spent on marketing to achieve the maximum profit is \( \boxed{100} \) thousand dollars.
\[
P = -5x^2 + 1000x + 5000
\]
This is a quadratic equation in the form \( P = ax^2 + bx + c \), where \( a = -5 \), \( b = 1000 \), and \( c = 5000 \).
The x-coordinate of the vertex of a parabola described by the equation \( ax^2 + bx + c \) is given by the formula:
\[
x = -\frac{b}{2a}
\]
Substituting the values of \( a \) and \( b \):
\[
x = -\frac{1000}{2 \times -5} = -\frac{1000}{-10} = 100
\]
Thus, the amount spent on marketing to achieve the maximum profit is \( \boxed{100} \) thousand dollars.