1. **Problem statement:** Given a cone with radius $r = x$ and height $h = 4x$, prove that its volume is $\frac{4}{3}\pi x^3$. 2. **Formula for volume of a cone:** $$V = \frac{1}{3}\pi r^2 h$$ This formula calculates the volume of a cone using its radius $r$ and height $h$. 3. **Substitute given values:** Since $r = x$ and $h = 4x$, substitute these into the formula: $$V = \frac{1}{3}\pi (x)^2 (4x)$$ 4. **Simplify the expression:** $$V = \frac{1}{3}\pi x^2 \times 4x = \frac{4}{3}\pi x^3$$ 5. **Conclusion:** We have shown that the volume of the cone is indeed: $$\boxed{\frac{4}{3}\pi x^3}$$ which is what we needed to prove.