Math Problem Statement

Find the maximum sum of two positive numbers (not necessarily integers), each of which is in [0, 400] whose product is 500.

Solution

To solve this problem, we need to maximize the sum of two positive numbers xx and yy, where xx and yy are in the interval [0,400][0, 400] and their product is 500. This translates into the following steps:

  1. Define the Relationship Between xx and yy: Since the product of xx and yy is 500, we have: xy=500x \cdot y = 500 Solving for yy, we get: y=500xy = \frac{500}{x}

  2. Express the Sum SS: We want to maximize the sum S=x+yS = x + y. Substituting y=500xy = \frac{500}{x}: S=x+500xS = x + \frac{500}{x}

  3. Find the Maximum of SS: To find the maximum of SS, we can differentiate it with respect to xx and set the derivative equal to zero.

    dSdx=1500x2\frac{dS}{dx} = 1 - \frac{500}{x^2} Setting dSdx=0\frac{dS}{dx} = 0: 1500x2=01 - \frac{500}{x^2} = 0 x2=500x^2 = 500 x=500=10522.36x = \sqrt{500} = 10\sqrt{5} \approx 22.36

  4. Calculate yy and the Sum SS: When x=105x = 10\sqrt{5}, y=500105=105y = \frac{500}{10\sqrt{5}} = 10\sqrt{5} as well. Therefore, both xx and yy are approximately 22.36.

    Thus, the maximum sum SS is: S=x+y=105+105=20544.72S = x + y = 10\sqrt{5} + 10\sqrt{5} = 20\sqrt{5} \approx 44.72

Answer:

The maximum sum of two positive numbers in the interval [0,400][0, 400] whose product is 500 is approximately 44.72.

Would you like a more detailed explanation, or do you have any specific questions?

Here are some related questions:

  1. How can we confirm that this value of xx and yy gives the maximum sum?
  2. What happens if the interval changes, say to [0,300][0, 300]?
  3. How do we solve a similar problem where the product constraint changes, e.g., xy=1000x \cdot y = 1000?
  4. What is the impact of changing the range of possible values for xx and yy?
  5. How do we handle maximization when there is more than one variable product constraint?

Tip: For optimization problems with constraints, expressing one variable in terms of the other is often a key step to simplifying and solving the problem.

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Math Problem Analysis

Mathematical Concepts

Optimization
Algebra

Formulas

Product of two numbers (x * y = 500)
Sum of two numbers (S = x + y)
Derivative of a function (dS/dx)

Theorems

Optimization using derivatives

Suitable Grade Level

Grades 11-12