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

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?

GPT-4o mini · Answer

The profit function given is $$ P = -5x^2 + 1000x + 5000 $$ This is a quadratic function in the standard form $ P = ax^2 + bx + c $, where $ a = -5 $, $ b = 1000 $, and $ c = 5000 $. Since the coefficient of $ x^2 $ (which is $ a $) is negative, this function opens downwards, indicating that there is a maximum profit. To find the value of $ x $ that maximizes $ P $, we can use the vertex formula for a quadratic equation. The $ x $-coordinate of the vertex, where the maximum occurs, is given by: $$ x = -\frac{b}{2a} $$ Substituting $ a = -5 $ and $ b = 1000 $ into the formula: $$ x = -\frac{1000}{2 \times -5} = -\frac{1000}{-10} = 100 $$ Thus, the amount spent on marketing to achieve maximum profit is $ x = 100 $ (in thousands of dollars). Therefore, the optimal marketing spending to achieve maximum profit is $$ \boxed{100} $$ (which corresponds to $100,000).