Question

The profit of a company is represented by P=-5x^2+1000x+5000 where P is profit and x is amount spent on marketing (in thousands of dollars). How much spending in thousands of dollars will be directed toward marketing to achieve maximum profit?

Answer 1

To find the amount spent on marketing that will achieve the maximum profit, we can analyze the quadratic profit function:

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

This is a quadratic function in the standard form \(P = ax^2 + bx + c\), where:
- \(a = -5\)
- \(b = 1000\)
- \(c = 5000\)

Since \(a < 0\), this parabola opens downwards, which means it has a maximum point.

The \(x\)-coordinate of the vertex of a parabola given by the function \(P = ax^2 + bx + c\) can be found using the formula:

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

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

\[
x = -\frac{1000}{2 \cdot -5} = -\frac{1000}{-10} = 100
\]

Thus, the maximum profit occurs when \(x = 100\).

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

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