The profit (in thousands of dollars) of a company is represented as P=−5x2+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?

Answer 1

To determine the amount spent on marketing that will achieve the maximum profit, we analyze the profit function given by:

\[ P = -5x^2 + 1000x + 5000 \]

This is a quadratic equation in the standard form \( P = ax^2 + bx + c \), where \( a = -5 \), \( b = 1000 \), and \( c = 5000 \). Since \( a < 0 \), the parabola opens downwards and has a maximum point at its vertex.

The x-coordinate of the vertex of a quadratic function can be calculated using the formula:

\[ x = -\frac{b}{2a} \]

Substituting the values of \( a \) and \( b \):

\[ x = -\frac{1000}{2 \times -5} = -\frac{1000}{-10} = 100 \]

Therefore, the amount spent on marketing to achieve the maximum profit is:

\[ \boxed{100} \text{ (in thousands of dollars)} \]