The problem states that y varies as sqrt(x). This can be written as:

The problem states that $y$ varies as $\sqrt{x}$. This can be written as:

$$y = k\sqrt{x}$$

where $k$ is the constant of proportionality.

We are given that the sum of the values of $y$ when $x=4$ and $x=100$ is 15.
When $x=4$, let $y_1 = k\sqrt{4} = k \times 2 = 2k$.
When $x=100$, let $y_2 = k\sqrt{100} = k \times 10 = 10k$.
The sum of these values is 15:

$$y_1 + y_2 = 15$$ $$2k + 10k = 15$$ $$12k = 15$$

Divide by 12 to find $k$:

$$k = \frac{15}{12}$$ $$k = \frac{5}{4}$$

Substitute the value of $k$ back into the variation equation from Step 1:

$$y = \frac{5}{4}\sqrt{x}$$

The law connecting $y$ and $x$ is $\boxed{y = \frac{5}{4}\sqrt{x}}$.