Surface Areas and Volumes | Exercise 12.1

Question 6

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.

Solution

We need to find the surface area of the capsule. The capsule is made of a cylinder and two hemispheres.

Step 1 — Find the dimensions

Let's first find the radius of the capsule. The diameter of the capsule is 5 mm. The radius is half of the diameter.

$r = \frac{5}{2}$

$\boxed{r = 2.5 \text{ mm}}$

This is the radius for both the cylinder and the hemispheres.

The total length of the capsule is 14 mm. Each hemisphere adds a length equal to its radius. So, two hemispheres add $2 \times \text{radius}$ to the length.

Length of cylindrical part = Total length - (radius + radius)

$h = 14 - (2.5 + 2.5)$

$h = 14 - 5$

$\boxed{9 \text{ mm}}$

This is the height of the cylindrical part.

Diagram 1

Step 2 — Calculate the surface area

The surface area of the capsule is the sum of three parts. It is the curved surface area of the cylinder. It is also the curved surface area of the two hemispheres.

Surface area of capsule = CSA of cylinder + (2 × CSA of hemisphere)

Curved Surface Area (CSA) of a cylinder = $2\pi rh$ . Curved Surface Area (CSA) of a hemisphere = $2\pi r^2$ .

Surface area of capsule $= 2\pi rh + 2(2\pi r^2)$

$= 2\pi rh + 4\pi r^2$

We can take $2\pi r$ common from both terms.

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

Now, let's substitute the values we found. Radius $r = 2.5 \text{ mm}$ . Height of cylinder $h = 9 \text{ mm}$ .

$= 2 \times \pi \times 2.5 \times (9 + 2 \times 2.5)$

$= 5\pi \times (9 + 5)$

$= 5\pi \times 14$

$= 70\pi$

We will use $\pi = \frac{22}{7}$ for the calculation.

$= 70 \times \frac{22}{7}$

$= 10 \times 22$

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

Answer

The surface area of the capsule is 220 mm².

Question 6

Answer

More questions in Exercise 12.1