1. **Problem statement:** We have a solid shape made up of a cone and a hemisphere. The radius of both the hemisphere and the cone is $x$ cm, and the perpendicular height of the cone is $2x$ cm. We need to find an expression for the total volume $V$ of the shape in terms of $x$. 2. **Formulas used:** - Volume of a cone: $$V_{cone} = \frac{1}{3} \pi r^2 h$$ - Volume of a hemisphere: $$V_{hemisphere} = \frac{2}{3} \pi r^3$$ 3. **Substitute the given values:** - Radius $r = x$ - Height of cone $h = 2x$ 4. **Calculate the volume of the cone:** $$V_{cone} = \frac{1}{3} \pi x^2 (2x) = \frac{2}{3} \pi x^3$$ 5. **Calculate the volume of the hemisphere:** $$V_{hemisphere} = \frac{2}{3} \pi x^3$$ 6. **Total volume $V$ is the sum of the volumes:** $$V = V_{cone} + V_{hemisphere} = \frac{2}{3} \pi x^3 + \frac{2}{3} \pi x^3 = \frac{4}{3} \pi x^3$$ **Final answer:** $$V = \frac{4}{3} \pi x^3$$