How to Find Maximum Profit Using Quadratic Functions
Quick Answer
To find the maximum profit from a quadratic profit function, use the vertex formula. For a function like P(l) = -4l² + 32l - 52, the vertex gives you the maximum profit value, which you can calculate even if L isn't provided directly.
Finding the maximum profit in a quadratic function may seem challenging, especially when a specific variable isn't provided. However, by understanding the properties of quadratic equations, you can easily find the solution. In this case, your profit function is given as P(l) = -4l² + 32l - 52. The key to solving this problem lies in finding the vertex of the parabola represented by this equation.
Quadratic functions are typically written in the form P(l) = al² + bl + c, where 'a', 'b', and 'c' are constants. For our function, 'a' is -4, 'b' is 32, and 'c' is -52. Since the coefficient of l² (which is -4) is negative, the parabola opens downwards, indicating that the maximum profit occurs at the vertex.
To find the vertex, you can use the formula for the l-coordinate of the vertex: l = -b/(2a). Substituting the values of a and b into this formula gives:
l = -32 / (2 * -4) = -32 / -8 = 4.
Now that we have the value of l, we can substitute it back into the original profit function to find the maximum profit:
P(4) = -4(4)² + 32(4) - 52.
Calculating this step-by-step:
1. First, calculate (4)² = 16.
2. Then, -4 * 16 = -64.
3. Next, 32 * 4 = 128.
4. Combine these results: -64 + 128 - 52 = 12.
Thus, the maximum profit is 12, which, as noted in the problem, is in thousands. Therefore, the maximum profit is $12,000.
This method is a fundamental technique in mathematics, particularly in business applications where profit maximization is crucial. Quadratic functions are commonly used in economics and finance to model various situations, such as profit and loss, revenue, and cost functions. Understanding how to find the vertex of these functions can provide valuable insights into optimizing outcomes.
In summary, even without a direct number for L, you can determine it using the vertex formula for quadratic functions. This problem exemplifies how algebra can be applied to real-world scenarios, reinforcing the importance of mastering these mathematical concepts.
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