The profits (in thousands of dollars) of a company is represented as P= -5x^2 +1,000x +5,000, where P represents the profit and x represent 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 to achieve the maximum profit, we will analyze the quadratic profit function given by:

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

This is a quadratic equation of the form \( P = ax^2 + bx + c \), where:

• \( a = -5 \)
• \( b = 1000 \)
• \( c = 5000 \)

For a quadratic equation, the x-coordinate of the vertex gives us the maximum (or minimum) point, depending on the orientation (concave up or down). Since \( a < 0 \) in this case (the parabola opens downwards), the vertex represents the maximum profit.

The x-coordinate of the vertex can be found using 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:

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

This means that the company should spend $100,000 on marketing to achieve the maximum profit.