Chapter 2, Problem 20P

Minimizing a Sum Find two positive numbers whose sum is 100 and the sum of whose squares is a minimum.

To determine

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

Answer to Problem 20P

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,

$x=-\frac{b}{2a}$ (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,

$y=100-x$ (2)

The sum of the squares of two numbers is,

$\begin{array}{c}S=x^2+{\left(100-x\right)}^2\\ =x^2+10,000+x^2-200x\\ =2x^2+10,000-200x\end{array}$

Now, compare the sum $S=2x^2+10,000-200x$ with the general equation of the quadratic equation $a{x}^2+bx+c=0$ then,

$a=2$

$b=-200$

And

$c=10,000$

The minimum function occurs at the point $x=-\frac{b}{2a}$.

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

$\begin{array}{c}x=-\frac{\left(-200\right)}{2\cdot 2}\\ =\frac{200}{4}\\ =50\end{array}$

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

$\begin{array}{c}y=100-50\\ =50\end{array}$

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