A vessel is in the form of a hollow hemisphere mounted by a hollow cylinder. The diameter of the hemisphere is and the total height of the vessel is . Find the inner surface area of the vessel.

Surface Areas and Volumes | Exercise 12.1

Question 2

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.

Solution

The vessel is made of two parts joined together — a cylinder at the top and a hemisphere (half of a sphere) at the bottom.

To find the inner surface area, we simply add the curved surface area (CSA) of each part.

Step 1 — Find the radius

The radius is always half of the diameter. So we use the formula:

$r = \frac{\text{diameter}}{2}$

Here the diameter is $14\text{ cm}$ , so:

$r = \frac{14}{2} = 7\text{ cm}$

Both the cylinder and the hemisphere share this same radius, $r = 7\text{ cm}$ .

Step 2 — Find the height of the cylinder

Look at the figure: the total height is made up of the cylinder's height plus the hemisphere's height.

For a hemisphere, its height is equal to its radius. So the hemisphere is $7\text{ cm}$ tall.

To get the cylinder's height $h$ , subtract the hemisphere's height from the total height:

$h = \text{total height} - \text{radius}$

$h = 13 - 7$

$\boxed{h = 6\text{ cm}}$

Diagram 1

Step 3 — Calculate the inner surface area

The inner surface we can touch is made of two curved parts:

the curved surface of the cylinder → CSA $= 2\pi rh$
the curved surface of the hemisphere → CSA $= 2\pi r^2$

Adding them together and taking $\pi = \dfrac{22}{7}$ :

$\text{Inner surface area} = 2\pi rh + 2\pi r^2$

Since $2\pi r$ is common to both terms, we can take it out to make the calculation easier:

$= 2\pi r\,(h + r)$

$= 2 \times \frac{22}{7} \times 7 \times (6 + 7)$

The $7$ in the top and the $7$ in the bottom cancel out:

$= 2 \times 22 \times 13$

$= 44 \times 13$

$\boxed{572\text{ cm}^2}$

Answer

The inner surface area of the vessel is $\mathbf{572\text{ cm}^2}$ .

Question 2

Answer

More questions in Exercise 12.1