The profit (in thousands of dollars) of a company is represented as P=−5x2+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?(1 point)
$
To find the amount spent on marketing that will result in maximum profit, we need to find the vertex of the parabola represented by the profit function \( P = -5x^2 + 1000x + 5000 \).
In a quadratic equation of the form \( P = ax^2 + bx + c \), where the coefficient \( a \) is negative (which indicates a downward opening parabola), the x-coordinate of the vertex can be found using the formula:
\[ x = \frac{-b}{2a} \]
In this case, \( a = -5 \) and \( b = 1000 \), so we can plug these values into the vertex formula:
\[ x = \frac{-1000}{2 \cdot (-5)} \]
\[ x = \frac{-1000}{-10} \]
\[ x = 100 \]
Therefore, the company should spend $100,000 on marketing to achieve the maximum profit, since the value of \( x \) is in thousands of dollars.