# Two positive numbers x, y have a sum of 20. What are their values if one number plus the square root of the other is a) as large as possible, b) as small as possible?

Oct 6, 2017

Maximum is $19 + \sqrt{1} = 20 \to$ $x = 19 , y = 1$

Minimum is $1 + \sqrt{19} = 1 + 4.36 = 5 \left(r o u n \mathrm{de} d\right) \to$$x = 1 , y = 19$

#### Explanation:

Given: $x + y = 20$

Find $x + \sqrt{y} = 20$ for max and min values of the sum of the two.

To obtain the max number, we would need to maximize the whole number and minimize the number under the square root:

That means: $x + \sqrt{y} = 20 \to 19 + \sqrt{1} = 20 \to \max$ [ANS]

To obtain the min number, we would need to minimize the whole number and maximize the number under the square root:

That is: $x + \sqrt{y} = 20 \to 1 + \sqrt{19} = 1 + 4.36 = 5 \left(r o u n \mathrm{de} d\right)$[ANS]