Q6:

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 $14$ mm and the diameter of the capsule is $5$ mm. Find its surface area.

Given:

• The shape of the capsule consists of a central cylinder and two hemispheres at each end.
• Total length of the capsule ($L$) = $14$ mm.
• Diameter of the capsule ($d$) = $5$ mm.

To Find:

The total surface area of the medicine capsule.

Step 1: Determine the dimensions of the cylinder and hemispheres.

The diameter of the capsule is $5$ mm, so the radius ($r$) of the cylinder and the hemispheres is:

$r = \frac{d}{2} = \frac{5}{2} = 2.5$ mm.

The length of the cylindrical part ($h$) is obtained by subtracting the radii of the two hemispheres from the total length of the capsule:

$h = L - (r + r) = 14 - (2.5 + 2.5) = 14 - 5 = 9$ mm.

Step 2: Identify the formula for the total surface area.

The total surface area of the capsule is the sum of the Curved Surface Area (CSA) of the cylinder and the Curved Surface Areas of the two hemispheres.

Total Surface Area = (CSA of Cylinder) + 2 $\times$ (CSA of Hemisphere)

Formulae:

• CSA of Cylinder = $2\pi rh$
• CSA of Hemisphere = $2\pi r^2$

Step 3: Calculate the surface area.

Total Surface Area = $2\pi rh + 2(2\pi r^2)$

Total Surface Area = $2\pi rh + 4\pi r^2$

Factor out $2\pi r$:

Total Surface Area = $2\pi r(h + 2r)$

Substitute the values ($r = 2.5$, $h = 9$, $\pi \approx \frac{22}{7}$):

Total Surface Area = $2 \times \frac{22}{7} \times 2.5 \times (9 + 2(2.5))$

Total Surface Area = $2 \times \frac{22}{7} \times 2.5 \times (9 + 5)$

Total Surface Area = $2 \times \frac{22}{7} \times 2.5 \times 14$

Step 4: Final Arithmetic Calculation.

Total Surface Area = $2 \times 22 \times 2.5 \times \frac{14}{7}$

Total Surface Area = $44 \times 2.5 \times 2$

Total Surface Area = $44 \times 5$

Total Surface Area = $220$ mm$^2$

Final Answer: The total surface area of the medicine capsule is 220 mm$^2$.