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

Question

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?(1 point)
$

Answer 1

To find the amount spent on marketing (represented by \( x \)) that achieves the maximum profit, we need to determine the vertex of the quadratic function given by:

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

The formula for the vertex \( x \) of a quadratic equation in the form \( ax^2 + bx + c \) is given by:

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

In this case, \( a = -5 \) and \( b = 1000 \). Plugging in these values, we get:

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

Thus, the amount of spending directed toward marketing to achieve maximum profit is:

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