Surface Areas and Volumes | Exercise 12.1

Question 9

A wooden article was made by scooping out a hemisphere from each end of a solid cylinder, as shown in Fig. 12.11. If the height of the cylinder is 10 cm, and its base is of radius 3.5 cm, find the total surface area of the article.

Solution

The total surface area is the sum of the curved surface area of the cylinder and the curved surface areas of the two hemispheres.

Step 1 — Note the dimensions

Let's write down the given measurements. The height of the cylinder is $\mathbf{10 \text{ cm}}$ . Its radius is $\mathbf{3.5 \text{ cm}}$ . The radius of each hemisphere is also $\mathbf{3.5 \text{ cm}}$ .

Diagram 1

Step 2 — Calculate cylinder's curved surface area

We will find the curved surface area of the cylinder. The formula is $2 \pi r h$ .

$2 \times \frac{22}{7} \times 3.5 \times 10$

$= 2 \times 22 \times 0.5 \times 10$

$= 220$

$\boxed{220 \text{ cm}^2}$

Step 3 — Calculate hemispheres' curved surface area

Now, let's find the curved surface area of the two hemispheres. The formula for one hemisphere is $2 \pi r^2$ . For two hemispheres, it is $2 \times (2 \pi r^2) = 4 \pi r^2$ .

$4 \times \frac{22}{7} \times (3.5)^2$

$= 4 \times \frac{22}{7} \times 3.5 \times 3.5$

$= 4 \times 22 \times 0.5 \times 3.5$

$= 154$

$\boxed{154 \text{ cm}^2}$

Step 4 — Find the total surface area

We add the curved surface areas from Step 2 and Step 3. This gives us the total surface area of the article.

$220 + 154$

$= 374$

$\boxed{374 \text{ cm}^2}$

Answer

(i) The total surface area of the article is $\mathbf{374 \text{ cm}^2}$ .

More questions in Exercise 12.1

Unless stated otherwise, take $\pi = \frac{22}{7}$ .

2 cubes each of volume $64\text{ cm}^3$ are joined end to end. Find the surface area of the resulting cuboid.

A vessel is in the form of a hollow hemisphere mounted by a hollow cylinder. The diameter of the hemisphere is $14\text{ cm}$ and the total height of the vessel is $13\text{ cm}$ . Find the inner surface area of the vessel.

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

A cubical block of side $7\text{ cm}$ is surmounted by a hemisphere. What is the greatest diameter the hemisphere can have? Find the surface area of the solid.

A hemispherical depression is cut out from one face of a cubical wooden block such that the diameter $l$ of the hemisphere is equal to the edge of the cube. Determine the surface area of the remaining solid.

A medicine capsule is in the shape of a cylinder with two hemispheres stuck to each of its ends (see Fig. 12.10). The length of the entire capsule is $14\text{ mm}$ and the diameter of the capsule is $5\text{ mm}$ . Find its surface area.

A tent is in the shape of a cylinder surmounted by a conical top. If the height and diameter of the cylindrical part are 2.1 m and 4 m respectively, and the slant height of the top is 2.8 m, find the area of the canvas used for making the tent. Also, find the cost of the canvas of the tent at the rate of ₹500 per m². (Note that the base of the tent will not be covered with canvas.)

From a solid cylinder whose height is 2.4 cm and diameter 1.4 cm, a conical cavity of the same height and same diameter is hollowed out. Find the total surface area of the remaining solid to the nearest cm².