The profit function P(x) of a firm, selling x items per day is given by P(x) = (150-x)x - 1625. Find the number of items the firm should manufacture to get maximum profit and find... The profit function P(x) of a firm, selling x items per day is given by P(x) = (150-x)x - 1625. Find the number of items the firm should manufacture to get maximum profit and find the maximum profit.
Understand the Problem
The question is asking to determine the number of items that a firm should manufacture in order to maximize profit, as well as to find out what that maximum profit amount is. We will approach this by finding the vertex of the quadratic profit function P(x) = (150-x)x - 1625, which represents a parabola and will allow us to determine the maximum point.
Answer
75 items, $4000
Answer for screen readers
The firm should manufacture 75 items to maximize profit, which is $4000.
Steps to Solve
- Identify the Profit Function
The given profit function is
$$ P(x) = (150 - x)x - 1625 $$
We can rewrite this in standard form by expanding it.
- Expand the Profit Function
Expand the equation to form a quadratic equation:
$$ P(x) = 150x - x^2 - 1625 $$
This can be rearranged to:
$$ P(x) = -x^2 + 150x - 1625 $$
- Find the Vertex of the Quadratic
The vertex of a parabola defined by the equation $ax^2 + bx + c$ can be found using the formula:
$$ x = -\frac{b}{2a} $$
Here, $a = -1$ and $b = 150$. Substituting these values:
$$ x = -\frac{150}{2 \cdot (-1)} = \frac{150}{2} = 75 $$
- Calculate Maximum Profit
Now, substitute $x = 75$ back into the profit function to find the maximum profit:
$$ P(75) = -75^2 + 150(75) - 1625 $$
Calculating it step-by-step:
- Calculate $75^2 = 5625$
- Calculate $150 \cdot 75 = 11250$
So,
$$ P(75) = -5625 + 11250 - 1625 $$
- Finally, combine these:
$$ P(75) = 11250 - 5625 - 1625 = 11250 - 7250 = 4000 $$
- Summarize the Findings
The maximum number of items to manufacture for maximum profit is 75, and the maximum profit is $4000.
The firm should manufacture 75 items to maximize profit, which is $4000.
More Information
The maximum profit occurs at the vertex of the parabolic profit function, and finding this point is critical in maximizing business profits. This method applies to various profit-related functions in economics.
Tips
- Forgetting to expand the profit function correctly.
- Miscalculating the vertex by not applying the formula $x = -\frac{b}{2a}$ accurately.
- Failing to substitute back into the profit function after finding $x$.
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