Quadratic Functions from Situations Practice

Question

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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)
$
Remaining Attempts : 3
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Answer 1

To determine the amount spent on marketing that will achieve maximum profit, we first need to identify the vertex of the quadratic function given by the equation \( P = -5x^2 + 1000x + 5000 \).

The general form of a quadratic function is \( ax^2 + bx + c \). Here, we have:

• \( a = -5 \)
• \( b = 1000 \)
• \( c = 5000 \)

The x-coordinate of the vertex (which gives the value of x that maximizes profit) can be calculated using the formula: \[ x = -\frac{b}{2a} \]

Plugging in the values of a and b: \[ x = -\frac{1000}{2 \times -5} = -\frac{1000}{-10} = 100 \]

So, the optimal spending directed toward marketing to achieve the maximum profit is \( 100 \) (in thousands of dollars).

To answer your question: $100,000 would be the amount spent on marketing to achieve the maximum profit.