Explanation:

- The volume of a cylinder is calculated using the formula: $V=\pi r^{2}h$.

- Given: The base radius ratio of the two cylinders is 3:4, meaning if the radius of the first cylinder is 3x, then the radius of the second is 4x.

- Similarly, the heights of the cylinders are in the ratio 4:9. If the height of the first cylinder is 4y, then the height of the second is 9y.

- To find the ratio of the volumes:

$$

\text{Volume of first cylinder} = \pi (3x)^2 (4y) = \pi \times 9x^2 \times 4y = 36\pi x^2y

$$

$$

\text{Volume of second cylinder} = \pi (4x)^2 (9y) = \pi \times 16x^2 \times 9y = 144\pi x^2y

$$

- The ratio of the volumes is:

$$

\frac{36\pi x^2y}{144\pi x^2y} = \frac{36}{144} = \frac{1}{4}

$$

- Correct Answer: Option 2: 1:4