The profit (in thousands of dollars) of a tech company is represented as

Question

P = -4x2 + 800x + 3,000, 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 find the amount spent on marketing that will achieve the maximum profit, we need to analyze the quadratic profit function given:

\[ P = -4x^2 + 800x + 3000 \]

This is a quadratic equation of the form \( P = ax^2 + bx + c \), where \( a = -4 \), \( b = 800 \), and \( c = 3000 \). Since \( a \) is negative, the parabola opens downward, meaning the vertex of the parabola will give us the maximum profit.

The x-coordinate of the vertex for a quadratic equation is calculated using the formula:

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

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

\[ x = -\frac{800}{2 \times -4} = -\frac{800}{-8} = 100 \]

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

\[ \boxed{100} \]

(meaning 100 thousand dollars).