# The two positive numbers whose sum is 100 and the sum of whose squares is a minimum.

### Precalculus: Mathematics for Calcu...

6th Edition
Stewart + 5 others
Publisher: Cengage Learning
ISBN: 9780840068071

### Precalculus: Mathematics for Calcu...

6th Edition
Stewart + 5 others
Publisher: Cengage Learning
ISBN: 9780840068071

#### Solutions

Chapter 2, Problem 20P
To determine

## To find: The two positive numbers whose sum is 100 and the sum of whose squares is a minimum.

Expert Solution

The two positive numbers are 50 and 50 whose sum is 100 and the sum of whose squares is a minimum.

### Explanation of Solution

Given:

The sum of the numbers is 100 and the sum of whose squares is a minimum.

Formula used:

The minimum of the function occurs at the point,

x=b2a (1)

Calculation:

Let the two numbers be x and y and the sum of whose squares is S.

The sum of the numbers represented as,

x+y=100

Now, get the value of y in terms of x,

y=100x (2)

The sum of the squares of two numbers is,

S=x2+(100x)2=x2+10,000+x2200x=2x2+10,000200x

Now, compare the sum S=2x2+10,000200x with the general equation of the quadratic equation ax2+bx+c=0 then,

a=2

b=200

And

c=10,000

The minimum function occurs at the point x=b2a .

Substitute 200 for b and 2 for a in the equation (1)

x=(200)22=2004=50

Now substitute 50 for x in the equation (2),

y=10050=50

Thus, the two numbers is 50 and 50 whose sum is 100 and the sum of whose squares is a minimum.

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