If the product of two positive numbers is 3600, what is the minimum value of their sum?
Overview
Given the problem, that if the product of two positive numbers is 3600, what is the minimum value of their sum. Let the numbers be a and b. Given that the product of the numbers is 3600.
ab = 3600
We can use the AM-GM Inequality here to solve the problem. The arithmetic mean-geometric mean (AM-GM) inequality states that the arithmetic mean of positive real numbers is greater than or equal to the geometric mean of the numbers.
AM-GM Inequality
Let a and b are positive numbers, arithmetic mean is greater than or equal to the geometric mean.
AM-GM Inequality states that:
or more in general for n numbers, the equation states that :
Now to answer the question: we known the product ab = 3600.
replace the product of the two numbers in the equation
we get:
The square root of 3600 is 60. So we get
Answer:
Hence the minimum value of the sum of the numbers is 120.
Proof
Proof for the AM-GM equation for two numbers:
Square both sides, we get
Expand and rearrange the terms, we get:
This equation holds true since the square of a number is always positive i.e
Let y = a-b