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)
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GPT-4o mini · Answer

To find the amount of spending in thousands that will achieve the maximum profit, we need to determine the vertex of the parabola described by the profit function $ P = -5x^2 + 1000x + 5000 $.

The vertex $ x $ can be found using the formula:

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

where $ a $ and $ b $ are the coefficients from the quadratic equation $ P = ax^2 + bx + c $.

From the given profit function:

$ a = -5 $
$ b = 1000 $

Now we can plug these values into the vertex formula:

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

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