The profit (in thousands of dollars) of a company is represented as P= -5a? + 1, 000z + 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? (1 point)

Answer 1

To find the amount spent on marketing that maximizes the profit \( P \), we start with the profit function given as:

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

This is a quadratic equation in the form \( P = -5a^2 + 1000a + 5000 \).

To determine the value of \( a \) that maximizes the profit, we use the vertex formula for a parabola, which is given by:

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

In our equation, the coefficients are:

• \( A = -5 \) (the coefficient of \( a^2 \))
• \( B = 1000 \) (the coefficient of \( a \))

Plugging in the values into the formula:

\[ a = -\frac{1000}{2(-5)} = -\frac{1000}{-10} = 100 \]

Therefore, the amount spent on marketing that achieves the maximum profit is:

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