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

Answer 1

To find the amount of spending on marketing that will achieve the maximum profit, we need to determine the vertex of the quadratic equation represented by \( P = -5x^2 + 1,000x + 5,000 \).

The vertex formula for the x-coordinate of a parabola represented by \( ax^2 + bx + c \) is given by:

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

In our equation:

• \( a = -5 \)
• \( b = 1,000 \)

Substituting these values into the vertex formula:

\[ x = -\frac{1,000}{2 \times -5} \] \[ x = -\frac{1,000}{-10} \] \[ x = 100 \]

Thus, the amount spent on marketing to achieve the maximum profit is 100 thousand dollars.