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 spent on marketing that will maximize the profit represented by the quadratic equation

\[ P = -5x^2 + 1,000x + 5,000, \]

we can use the formula for the vertex of a quadratic equation in the standard form \( ax^2 + bx + c \). The x-coordinate of the vertex, which gives the maximum (or minimum) value of the quadratic, is calculated using the formula:

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

Here, \( a = -5 \) and \( b = 1,000 \).

Substituting in these values:

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

Thus, the maximum profit is achieved when \( x = 100 \) (thousands of dollars spent on marketing).

Therefore, the amount to be directed toward marketing to achieve the maximum profit is

\[ \boxed{100} \text{ thousand dollars.} \]