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)
$

Answer 1

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.