Math Problem Statement
Solution
To solve this problem, we need to maximize the sum of two positive numbers and , where and are in the interval and their product is 500. This translates into the following steps:
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Define the Relationship Between and : Since the product of and is 500, we have: Solving for , we get:
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Express the Sum : We want to maximize the sum . Substituting :
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Find the Maximum of : To find the maximum of , we can differentiate it with respect to and set the derivative equal to zero.
Setting :
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Calculate and the Sum : When , as well. Therefore, both and are approximately 22.36.
Thus, the maximum sum is:
Answer:
The maximum sum of two positive numbers in the interval 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:
- How can we confirm that this value of and gives the maximum sum?
- What happens if the interval changes, say to ?
- How do we solve a similar problem where the product constraint changes, e.g., ?
- What is the impact of changing the range of possible values for and ?
- 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
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