Mensuration: Surface Areas and Volumes - Compute surface areas of combinations of cubes, cuboids, cylinders, cones, spheres, and hemispheres

A toy is in the form of a cone of radius $3.5$ cm mounted on a hemisphere of the same radius. The total height of the toy is $15.5$ cm. Find the total surface area of the toy.

1. Radius of cone and hemisphere ($r$) = $3.5$ cm.
2. Total height of the toy = $15.5$ cm.
3. Height of the conical part ($h$) = Total height - Radius of hemisphere = $15.5 - 3.5 = 12$ cm.
4. Slant height of the cone ($l$): $l = \sqrt{r^2 + h^2} = \sqrt{(3.5)^2 + 12^2} = \sqrt{12.25 + 144} = \sqrt{156.25} = 12.5\text{ cm}$
5. TSA of toy = CSA of cone + CSA of hemisphere
6. TSA = $\pi rl + 2\pi r^2 = \pi r(l + 2r)$
7. TSA = $\frac{22}{7} \times 3.5 \times (12.5 + 2 \times 3.5)$
8. TSA = $11 \times (12.5 + 7) = 11 \times 19.5 = 214.5\text{ cm}^2$.

To solve this, we first identify that the flat circular faces of the cone and hemisphere are joined and hidden. Thus, we only add the curved surface areas. We must subtract the hemisphere's radius from the total height to find the cone's vertical height, then use Pythagoras to find the slant height.

A medicine capsule is in the shape of a cylinder with two hemispheres stuck to each of its ends. The length of the entire capsule is $14$ mm and the diameter of the capsule is $5$ mm. Find its surface area.

1. Diameter = $5$ mm, so Radius ($r$) = $2.5$ mm.
2. Total length = $14$ mm.
3. Length of cylindrical part ($h$) = Total length - (Radius of left hemisphere + Radius of right hemisphere) = $14 - (2.5 + 2.5) = 9$ mm.
4. Surface area of capsule = CSA of cylinder + $2 \times$ CSA of hemisphere
5. SA = $2\pi rh + 2(2\pi r^2) = 2\pi rh + 4\pi r^2 = 2\pi r(h + 2r)$
6. SA = $2 \times \frac{22}{7} \times 2.5 \times (9 + 2 \times 2.5)$
7. SA = $2 \times \frac{22}{7} \times 2.5 \times 14$
8. SA = $22 \times 2.5 \times 4 = 220\text{ mm}^2$.

The capsule is a combination of one cylinder and two hemispheres. Since the hemispheres are at the ends, the total surface area is the sum of the curved surface area of the cylinder and the curved surface areas of both hemispheres. We calculate the cylinder's height by subtracting the radii of the two ends from the total length.

Given: Radius of hemisphere ($r$) = $1$ cm Radius of cone ($r$) = $1$ cm Height of cone ($h$) = $1$ cm

1. Calculate slant height ($l$) of the cone: $l = \sqrt{r^2 + h^2} = \sqrt{1^2 + 1^2} = \sqrt{2} \text{ cm}$

2. Surface Area of the solid = $CSA_{cone} + CSA_{hemisphere}$ $SA = \pi rl + 2\pi r^2$ $SA = \pi(1)(\sqrt{2}) + 2\pi(1)^2$ $SA = \pi\sqrt{2} + 2\pi$ $SA = \pi(2 + \sqrt{2}) \text{ cm}^2$

The surface area consists of the curved portion of the cone and the curved portion of the hemisphere. The circular bases where they meet are internal and not counted.

Given: Edge of cube ($a$) = $5$ cm Diameter of hemisphere = $4.2$ cm $\implies$ Radius ($r$) = $2.1$ cm

1. Surface area of cube = $6a^2 = 6(5)^2 = 150 \text{ cm}^2$

2. Area of the base of the hemisphere = $\pi r^2 = \frac{22}{7} \times 2.1 \times 2.1 = 13.86 \text{ cm}^2$

3. CSA of hemisphere = $2\pi r^2 = 2 \times 13.86 = 27.72 \text{ cm}^2$

4. Total Surface Area = (TSA of cube) - (Base area of hemisphere) + (CSA of hemisphere) $TSA = 150 - 13.86 + 27.72$ $\begin{array}{r} 150.00 \\ - 13.86 \\ \hline 136.14 \end{array}$ $\begin{array}{r} 136.14 \\ + 27.72 \\ \hline 163.86 \end{array}$ Total Surface Area = $163.86 \text{ cm}^2$

The hemisphere covers a part of the top face of the cube. We subtract the area of the circular base of the hemisphere from the cube's total surface area and then add the curved surface area of the hemisphere.