Solution: Volume of hemisphere: $\frac{2}{3}\pi (3x)^3 = \frac{2}{3}\pi \cdot 27x^3 = 18\pi x^3$.

["Solution: Volume of a Hemisphere Explained – Step-by-Step Formula and Calculation", "Understanding the volume of a hemisphere is essential in geometry, especially when solving problems related to spherical shapes in real-world applications like engineering, architecture, and physics. In this article, we explore the formula and step-by-step solution for finding the volume of a hemisphere with radius $3x$, showing how to derive the expression $\frac{2}{3}\pi (3x)^3 = 18\pi x^3$.", "### What is a Hemisphere?", "A hemisphere is half of a full sphere. Since the volume of a full sphere is given by the formula:", "$$\nV = \frac{4}{3}\pi r^3,\n$$", "the volume of a hemisphere is simply half of this:", "$$\nV_{\ ext{hemisphere}} = \frac{1}{2} \cdot \frac{4}{3}\pi r^3 = \frac{2}{3}\pi r^3.\n$$", "### Applying the Formula to Radius $3x$", "Now, let’s compute the volume when the radius $r = 3x$. Substitute $3x$ into the hemisphere volume formula:", "$$\nV = \frac{2}{3}\pi (3x)^3.\n$$", "Next, calculate $(3x)^3$:", "$$\n(3x)^3 = 3^3 \cdot x^3 = 27x^3.\n$$", "Now plug this back into the volume expression:", "$$\nV = \frac{2}{3}\pi \cdot 27x^3.\n$$", "Multiply the constants:", "$$\nV = \frac{2}{3} \cdot 27 \cdot \pi x^3 = 18\pi x^3.\n$$", "### Final Answer", "Thus, the volume of a hemisphere with radius $3x$ is:", "$$\n\boxed{18\pi x^3}\n$$", "This clear and structured approach makes it easy to apply the formula in future problems involving hemispheres. Whether you're calculating the volume of a storage tank, a dome, or modeling spherical objects in physics, knowing how to compute a hemisphere’s volume is key. Use this solution as a foundation for mastering three-dimensional geometry!"]