Question

The profit (in thousands of dollars) of a company is represented as P=−5x2+1,000x+5,000
𝑃
=
−
5
𝑥
2
+
1
,
000
𝑥
+
5
,
000
, where P
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

answer

Answer 2

The profit function given is:

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

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

- \( a = -5 \)
- \( b = 1,000 \)
- \( c = 5,000 \)

To find the value of \( x \) that will maximize the profit, we can use the vertex formula for a quadratic equation. The x-coordinate of the vertex (maximum point for a concave down parabola) is given by:

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

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

\[ x = -\frac{1,000}{2(-5)} \]

Calculating this:

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

Thus, the amount spent on marketing to achieve maximum profit is \( \boxed{100} \) (in thousands of dollars).

Answer 3

The amount spent on marketing to achieve maximum profit is \( \boxed{100} \) (in thousands of dollars).