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)

Question

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)

GPT-4o mini · Answer

To find the value of $ x $ that maximizes the profit $ P = -5x^2 + 1000x + 5000 $, we can use the vertex formula for a quadratic equation of the form $ ax^2 + bx + c $. The $ x $-coordinate of the vertex (which gives us the maximum profit since the parabola opens downward) can be calculated using the formula: $$ x = -\frac{b}{2a} $$ In the given equation, $ a = -5 $ and $ b = 1000 $. Substituting these values into the vertex formula, we get: $$ x = -\frac{1000}{2(-5)} = -\frac{1000}{-10} = 100 $$ Thus, the amount to be spent on marketing to achieve maximum profit is $ \boxed{100} $ (in thousands of dollars).