Optimizing the Sum of Squares with Two Numbers
In this article, we delve into a mathematical problem where we aim to find two numbers whose sum is 32, and the sum of their squares is minimized. This is a classic example of mathematical optimization and involves the application of quadratic equations. We will explore the process step-by-step and discuss the key concepts involved.
Introduction to the Problem
We are tasked with finding two numbers, denoted as ( x ) and ( y ), such that:
( x y 32 ) The sum of their squares, ( x^2 y^2 ), is minimized.To solve this, we represent ( y ) in terms of ( x ) using the first equation and then substitute it into the second equation.
Solution Process
Step 1: Express ( y ) in terms of ( x )
Given ( x y 32 ), we can express ( y ) as:
( y 32 - x )
Step 2: Substitute ( y ) into the sum of squares equation
We substitute ( y 32 - x ) into ( x^2 y^2 ), giving us:
( x^2 (32 - x)^2 )
Expanding this expression, we get:
( x^2 (32 - x)^2 x^2 32^2 - 64x x^2 )
Simplifying further:
( x^2 x^2 1024 - 64x 2x^2 - 64x 1024 )
Step 3: Finding the Minimum Value
To find the minimum value of the expression ( 2x^2 - 64x 1024 ), we can use the vertex formula for a quadratic equation, ( ax^2 bx c ), where the vertex occurs at ( x -frac{b}{2a} ).
In this case, ( a 2 ) and ( b -64 ), so:
( x -frac{-64}{2 cdot 2} frac{64}{4} 16 )
Step 4: Finding ( y )
Substituting ( x 16 ) back into the equation ( y 32 - x ), we get:
( y 32 - 16 16 )
Step 5: Calculating the Sum of the Two Numbers
The sum of the two numbers ( x ) and ( y ) is:
( x y 16 16 32 )
Hence, both numbers are 16, and their sum is 32, which minimizes the sum of their squares.
Alternative Approaches
Let's consider another approach to solve the problem:
In this approach, if one number is ( x ), then the other number is ( 32 - x ), and the sum of their squares is:
( f(x) x^2 (32 - x)^2 2x^2 - 64x 1024 )
The derivative of ( f(x) ) is:
( frac{d}{dx} (2x^2 - 64x 1024) 4x - 64 )
Setting the derivative to zero to find the critical points:
( 4x - 64 0 )
( x 16 )
Substituting ( x 16 ), we find ( f(16) 512 ), which is the minimum value of the sum of squares.
Conclusion
The problem of finding two numbers whose sum is 32 and whose sum of squares is minimized is solved by setting ( x ) and ( y ) equal to 16. The sum of these numbers is 32, and the minimum sum of their squares is 512. This approach demonstrates the application of quadratic equations and the concept of optimization in mathematics.