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)
$
Answers
GPT-4o mini
To find the amount spent on marketing that achieves the maximum profit, we need to determine the vertex of the quadratic function given by
\[
P = -5x^2 + 1,000x + 5,000
\]
For a quadratic function in the standard form \(P = ax^2 + bx + c\), the x-coordinate of the vertex, which gives the maximum (or minimum) value, can be found using the formula:
\[
x = -\frac{b}{2a}
\]
In this case, \(a = -5\) and \(b = 1,000\). Plugging in these values:
\[
x = -\frac{1,000}{2(-5)} = -\frac{1,000}{-10} = 100
\]
So, the amount directed toward marketing to achieve the maximum profit is \(x = 100\) thousands of dollars.
Thus, the answer is:
\[
\text{\$} 100,000
\]
\[
P = -5x^2 + 1,000x + 5,000
\]
For a quadratic function in the standard form \(P = ax^2 + bx + c\), the x-coordinate of the vertex, which gives the maximum (or minimum) value, can be found using the formula:
\[
x = -\frac{b}{2a}
\]
In this case, \(a = -5\) and \(b = 1,000\). Plugging in these values:
\[
x = -\frac{1,000}{2(-5)} = -\frac{1,000}{-10} = 100
\]
So, the amount directed toward marketing to achieve the maximum profit is \(x = 100\) thousands of dollars.
Thus, the answer is:
\[
\text{\$} 100,000
\]