This problem involves optimizing the product of two numbers under the constraint that their sum is related to another variable. Let's solve it step by step:

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Step 1: Define the variables

Let the two numbers be $x$ (the first number) and $y$ (the second number).

From the problem:

• Sum constraint: $x + 2y = 100$
• Objective: Maximize the product $P = xy$.

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Step 2: Express $x$ in terms of $y$

From the constraint $x + 2y = 100$, solve for $x$: $x = 100 - 2y$

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Step 3: Write the product in terms of $y$

The product $P = xy$ becomes: $P = (100 - 2y)y$ Simplify: $P = 100y - 2y^2$

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Step 4: Maximize the product

To maximize $P$, take its derivative with respect to $y$ and set it to zero: $\frac{dP}{dy} = 100 - 4y$ Set $\frac{dP}{dy} = 0$: $100 - 4y = 0 \implies y = 25$

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Step 5: Find $x$ and verify the maximum

Using $x = 100 - 2y$, substitute $y = 25$: $x = 100 - 2(25) = 50$

The numbers are $x = 50$ and $y = 25$.

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Step 6: Verify the maximum

The second derivative of $P$ is: $\frac{d^2P}{dy^2} = -4$ Since $\frac{d^2P}{dy^2} < 0$, the function $P$ has a maximum at $y = 25$.

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Final Answer:

The numbers are $x = 50$ and $y = 25$. The maximum product is: $P = 50 \cdot 25 = 1250$

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Would you like a detailed explanation of any step?

Here are 5 related questions to explore further:

1. How do you determine whether a critical point is a maximum or minimum?
2. What is the significance of the second derivative test in optimization problems?
3. How would the solution change if the constraint was $x + 3y = 100$?
4. Can you generalize this method for different constraints?
5. How is this method applied in real-world problems, like economics or engineering?

Tip: When solving optimization problems, always verify that your solution satisfies the original constraint!