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Find two positive numbers that satisfy the given requirements. (Enter your answers as a comma-separated list.)
The sum of the first and twice the second is 80 and the product is a maximum.
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Ask by Carter Ruiz. in the United States
Mar 24,2025

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40, 20

Solución

Let the two positive numbers be and where is the first number and is the second number.
  1. The condition given is:
    Solve for :
  2. The product of the two numbers is:
  3. To maximize , take the derivative of with respect to and set it equal to zero:
    Solving for :
  4. Substitute back into the equation for :
Thus, the two positive numbers are and .
40, 20

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Extra Insights

To solve this problem, we can define the two positive numbers as and . The conditions given are:
  2. We want to maximize the product .
From the first equation, we can express in terms of :
.
Now we substitute into the product equation:
.
To find the maximum product, we take the derivative of with respect to , set it to zero, and solve for :
Substituting back into the equation for :
Thus, the two numbers are and .
So, your answer is: .

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