A sphere, cylinder and cone of dimensions radius = $r$ cm and height = $2r$ cm are made. Which one has the greatest volume?

### Concept & Volume Formulas To compare the volumes, express the volume of each 3D shape strictly in terms of the variable $r$ using standard geometric formulas: $$ V_{\text{sphere}} = \frac{4}{3} \pi r^3 $$ $$ V_{\text{cylinder}} = \pi r^2 h $$ $$ V_{\text{cone}} = \frac{1}{3} \pi r^2 h $$ ### Step-by-Step Solution - The radius for all shapes is $r$. The height $h$ for the cylinder and cone is given as $2r$. - Calculate the volume of the Sphere: $V_s = \frac{4}{3} \pi r^3$. - Calculate the volume of the Cylinder: Substitute $h = 2r$ into the formula. $V_{cy} = \pi r^2 (2r) = 2 \pi r^3$. - Calculate the volume of the Cone: Substitute $h = 2r$ into the formula. $V_{co} = \frac{1}{3} \pi r^2 (2r) = \frac{2}{3} \pi r^3$. - Now, compare the coefficients of $\pi r^3$: - Cone: $\frac{2}{3}$ (approx $0.67$) - Sphere: $\frac{4}{3}$ (approx $1.33$) - Cylinder: $2$ - Comparing the values: $\frac{2}{3} < \frac{4}{3} < 2$. The cylinder has the largest coefficient. ### Exam Strategy & Shortcut Visualize the containment. A sphere of radius $r$ perfectly fits inside a cylinder of radius $r$ and height $2r$. Therefore, the cylinder must have a larger volume than the sphere. Similarly, a cone of the same dimensions is just $1/3$ the volume of the cylinder. Thus, the cylinder is clearly the largest. ### Common Pitfall Forgetting the specific formulas or failing to substitute $h = 2r$ accurately, leading to improper comparison of the coefficients. ### Final Answer Therefore, the correct answer is **Cylinder**.