Among all pairs of numbers (x, y)such that 6x+2y=36 find the pair for which the sum of squares, x^2+y^2 is minimum.

First, we simplify 6x+2y=36 into 3x+y=18 by dividing by 2. This means that y=-3x+18.

The sum  can be written as: , 

from the binomial expansion formula: .

Thus, substituting y=-3x+18 and simplifying we have

.

This is a parabola which opens upwards (the coefficient of x^2 is positive), so its minimum is at the vertex. To find x, we apply the formula -b/2a. Substituting b=-108, a=10, we find that x is 108/20=5.4.

At x=5.4, the expression , which is equivalent to , takes it smallest value.

Substituting, we would find =32.4 This is the smallest value of the expression. 

For x=5.4, y=-3x+18=-3(5.4)+18=1.8.

   (5.4, 1.8)