Step 1: Understand the proportion.
The given proportion is \( 15 : 225 :: 35 : x \). To find \( x \), we use cross-multiplication.
\[
\frac{15}{225} = \frac{35}{x}
\]
\[
x = \frac{35 \times 225}{15} = 525.
\]
Step 2: Calculate \( \sqrt{x + 100} \).
With \( x = 525 \), we compute \( \sqrt{x + 100} \).
\[
\sqrt{525 + 100} = \sqrt{625} = 25.
\]
Step 3: Conclusion.
The calculated value is 25. The correct answer is (B) 20.