Before solving the problem, I note the following:
Lemma: If n non-negative terms sum up to k, then the sum of their square roots is maximized if each term = k/n. The closer the individual terms are to the average, the larger the sum of the square. I could prove this with calculus, but will instead just use and example. Let n = 2 and k = 5.
sqrt(0) + sqrt(5) = 2.23
sqrt(1) + sqrt(4) = 3.00
sqrt(2) + sqrt(3) = 3.15
sqrt(2.5) + sqrt(2.5) = 3.16
So, what does this have to do with our problem?
Well, consider the sum of the square roots of a,b,c and d where a+b+c+d<=30. Clearly, the sum of the square roots is maximized if a+b+c+d=30. But a+b+c<= 14. The sum of the squares is therefore maximized if d = 30 - 14 = 16 and a+b+c = 14. It is for instance possible for d = 17 and a+b+c = 13, but this is necessarily a smaller sum of square roots (using our lemma).
Furthermore, using the same logic, if a+b+c = 14 and a+b<=9, then the sum of the square roots of a,b,c is maximized if c = 14 - 5 = 9 and a+b = 5
Furthermore, using the same logic, if a+b = 5 and a<=1, then the sum of the square roots of a,b is maximized if b = 5 - 1 = 4 and a = 1
So, the sum of the square roots is maximized if a = 1, b = 4, c = 9, and d = 16. And the sum of those square roots is 1+2+3+4 = 10.
So, finally, for any other values of a,b,c,d the sum of their square roots is necessarily less than or equal to the maximum sum of the square roots, which is 10.