Question
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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The sum of the first and twice the second is 80 and the product is a maximum.
Road It
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SUBMIT ANSWER
Ask by Carter Ruiz. in the United States
Mar 24,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
40, 20
Solution
Let the two positive numbers be
-
The condition given is:Solve for
: -
The product of the two numbers is:
-
To maximize
, take the derivative of with respect to and set it equal to zero: Solving for: -
Substitute
back into the equation for :
Thus, the two positive numbers are
40, 20
Answered by UpStudy AI and reviewed by a Professional Tutor
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Simplify this solution Extra Insights
To solve this problem, we can define the two positive numbers as
-
- We want to maximize the product
.
From the first equation, we can express
Now we substitute
To find the maximum product, we take the derivative of
Substituting
Thus, the two numbers are
So, your answer is:
