Mensuration: Surface Areas and Volumes - Compute volumes of composite solids formed from two standard three-dimensional shapes

A solid is in the form of a cone standing on a hemisphere with both their radii being equal to $1 \text{ cm}$ and the height of the cone is equal to its radius. Find the volume of the solid in terms of $\pi$.

Step 1: Identify the two solids. The object is a combination of a cone and a hemisphere. \nStep 2: List the given dimensions. Radius ($r$) of both cone and hemisphere = $1 \text{ cm}$. Height ($h$) of the cone = $1 \text{ cm}$. \nStep 3: Calculate the volume of the cone part: $V_1 = \frac{1}{3} \pi r^2 h = \frac{1}{3} \pi (1)^2 (1) = \frac{1}{3} \pi \text{ cm}^3$. \nStep 4: Calculate the volume of the hemisphere part: $V_2 = \frac{2}{3} \pi r^3 = \frac{2}{3} \pi (1)^3 = \frac{2}{3} \pi \text{ cm}^3$. \nStep 5: Add the volumes for the total volume: $V = V_1 + V_2 = \frac{1}{3} \pi + \frac{2}{3} \pi = \frac{3}{3} \pi = \pi \text{ cm}^3$.

Since the cone and hemisphere are joined together, we simply calculate their individual volumes using the standard formulae and add them. The problem asks for the answer 'in terms of $\pi$', so we do not substitute $3.14$ or $\frac{22}{7}$.

A decorative block is made of two solids — a cube and a hemisphere. The base of the block is a cube with edge $5 \text{ cm}$, and the hemisphere fixed on the top has a diameter of $4.2 \text{ cm}$. Find the total volume of the block. (Use $\pi = \frac{22}{7}$)

Step 1: Identify the parts. We have a cube and a hemisphere sitting on top of it. \nStep 2: Note the dimensions. Edge of cube ($a$) = $5 \text{ cm}$. Radius of hemisphere ($r$) = $\frac{4.2}{2} = 2.1 \text{ cm}$. \nStep 3: Calculate the volume of the cube: $V_{\text{cube}} = a^3 = 5^3 = 125 \text{ cm}^3$. \nStep 4: Calculate the volume of the hemisphere: $V_{\text{hemi}} = \frac{2}{3} \pi r^3 = \frac{2}{3} \times \frac{22}{7} \times (2.1)^3$. $V_{\text{hemi}} = \frac{2}{3} \times \frac{22}{7} \times 9.261 = 19.404 \text{ cm}^3$. \nStep 5: Total volume of the block = $V_{\text{cube}} + V_{\text{hemi}} = 125 + 19.404 = 144.404 \text{ cm}^3$.

In this problem, the hemisphere is placed on top of the cube. The volume of the block is the sum of the space occupied by the cube and the space occupied by the hemisphere. Note that for volume, the area where they touch (the base of the hemisphere) does not need to be subtracted, unlike when calculating surface area.

1. Radius of the cylinder and hemispheres $r = \frac{7}{2} = 3.5 \text{ cm}$.
2. Total length of toy $= 20 \text{ cm}$.
3. Height of the cylindrical part $h = 20 - (3.5 + 3.5) = 20 - 7 = 13 \text{ cm}$.
4. Volume of the toy $= \text{Volume of cylinder} + 2 \times \text{Volume of hemisphere}$ $V = \pi r^2 h + 2 \times \frac{2}{3} \pi r^3$ $V = \pi r^2 (h + \frac{4}{3} r)$ $V = \frac{22}{7} \times \frac{7}{2} \times \frac{7}{2} \times (13 + \frac{4}{3} \times \frac{7}{2})$ $V = \frac{11 \times 7}{2} \times (13 + \frac{14}{3})$ $V = 38.5 \times \frac{39 + 14}{3} = 38.5 \times \frac{53}{3}$ $V = 680.17 \text{ cm}^3$

To find the volume of a capsule-shaped solid, we calculate the volume of the central cylinder and add the volumes of the two hemispheres at the ends. Note that the height of the cylinder is found by subtracting the radii of both hemispheres from the total length.

1. Radius of cylinder and cone $r = 6 \text{ cm}$.
2. Height of cylinder $H = 8 \text{ cm}$.
3. Height of cone $h = 8 \text{ cm}$.
4. Total Volume $V = \text{Volume of Cylinder} + \text{Volume of Cone}$ $V = \pi r^2 H + \frac{1}{3} \pi r^2 h$ $V = \pi r^2 (H + \frac{1}{3} h)$ $V = \pi \times 6^2 \times (8 + \frac{8}{3})$ $V = 36\pi \times (\frac{24 + 8}{3})$ $V = 36\pi \times \frac{32}{3} = 12\pi \times 32$ $V = 384\pi \text{ cm}^3$ Taking $\pi \approx 3.14$: $V \approx 1205.76 \text{ cm}^3$

The volume of the composite solid is the sum of the volume of the cylinder and the volume of the cone. Since both share the same base, the radius is constant for both parts.