What is the Maximum Profit from the Function P(L) for Lip Gloss Production?
Quick Answer
To find the maximum profit for the function P(L) = -4L² + 12L + 25, calculate the vertex using L = -b/(2a). This gives L = 1.5, leading to a maximum profit of $34,000.
The function P(L) = -4L² + 12L + 25 represents the profit (in thousands of dollars) from producing L units of lip gloss. To determine the maximum profit, we need to identify the vertex of the quadratic equation, as it represents the highest point of the profit curve.
### Understanding the Quadratic Function
A quadratic function is generally in the form of P(L) = aL² + bL + c, where:
- **a** is the coefficient of L²,
- **b** is the coefficient of L,
- **c** is the constant term.
In our case:
- a = -4 (which indicates that the parabola opens downwards),
- b = 12,
- c = 25.
### Finding the Vertex
To find the maximum profit, we use the vertex formula for the x-coordinate of a parabola, which is given by:
L = -b / (2a)
Substituting our values of a and b:
- L = -12 / (2 * -4) = -12 / -8 = 1.5.
### Calculating Maximum Profit
Now that we have L = 1.5, we will substitute this value back into the profit function P(L) to find the maximum profit:
P(1.5) = -4(1.5)² + 12(1.5) + 25.
Calculating step-by-step:
1. Calculate (1.5)² = 2.25,
2. Multiply by -4: -4 * 2.25 = -9,
3. Calculate 12 * 1.5 = 18,
4. Finally, substitute back: P(1.5) = -9 + 18 + 25 = 34.
Thus, P(1.5) = 34 thousand dollars, or $34,000.
### Conclusion
The maximum profit when producing lip gloss is $34,000. Understanding how to work with quadratic functions is crucial, as they can be found in many real-world scenarios, such as maximizing profits in business, optimizing areas in geometry, and even in physics problems involving projectile motion. By mastering these concepts, you can apply them to various fields and scenarios effectively.
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