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Areas Related To Circles
EXERCISE 12.1
CBSE_NCERT_TEXTBOOK
Grade 10
Question:
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. Fig. 12.10 Fig. 12.9 SURFACE AREAS AND VOLUMES 167
Step-by-Step Solution
Key Concept: The capsule consists of a cylindrical part and two hemispherical ends which together form a sphere. Hence, total surface area = lateral surface area of the cylinder + surface area of the sphere. Use $\text{Lateral area of cylinder}=2\pi r h$ and $\text{Surface area of sphere}=4\pi r^{2}$.
1. Identify the dimensions\
- Diameter $=5\,\text{mm}\;\Rightarrow\;$ radius $r=\dfrac{5}{2}=2.5\,\text{mm}$.\
- Total length of capsule $=14\,\text{mm}$.\
- The two hemispherical ends together contribute a length of $2r = 5\,\text{mm}$.\
- Hence the length of the cylindrical part (height $h$) is\
$$h = 14\,\text{mm} - 2r = 14 - 5 = 9\,\text{mm}.$$\
2. Surface area of the cylindrical part\
$$\text{Lateral area of cylinder}=2\pi r h = 2\pi (2.5)(9) = 45\pi\,\text{mm}^2.$$\
3. Surface area of the two hemispheres\
Two hemispheres make a complete sphere, whose surface area is\
$$\text{Surface area of sphere}=4\pi r^{2}=4\pi (2.5)^{2}=4\pi (6.25)=25\pi\,\text{mm}^2.$$\
4. Total surface area of the capsule\
$$\text{Total SA}=\text{Lateral area of cylinder}+\text{Surface area of sphere}
=45\pi+25\pi = 70\pi\,\text{mm}^2.$$\
5. Numerical value (optional)\
$$70\pi \approx 70 \times 3.14 = 219.8\,\text{mm}^2 \;\text{(≈ 220 mm}^2\text{)}.$$
- Diameter $=5\,\text{mm}\;\Rightarrow\;$ radius $r=\dfrac{5}{2}=2.5\,\text{mm}$.\
- Total length of capsule $=14\,\text{mm}$.\
- The two hemispherical ends together contribute a length of $2r = 5\,\text{mm}$.\
- Hence the length of the cylindrical part (height $h$) is\
$$h = 14\,\text{mm} - 2r = 14 - 5 = 9\,\text{mm}.$$\
2. Surface area of the cylindrical part\
$$\text{Lateral area of cylinder}=2\pi r h = 2\pi (2.5)(9) = 45\pi\,\text{mm}^2.$$\
3. Surface area of the two hemispheres\
Two hemispheres make a complete sphere, whose surface area is\
$$\text{Surface area of sphere}=4\pi r^{2}=4\pi (2.5)^{2}=4\pi (6.25)=25\pi\,\text{mm}^2.$$\
4. Total surface area of the capsule\
$$\text{Total SA}=\text{Lateral area of cylinder}+\text{Surface area of sphere}
=45\pi+25\pi = 70\pi\,\text{mm}^2.$$\
5. Numerical value (optional)\
$$70\pi \approx 70 \times 3.14 = 219.8\,\text{mm}^2 \;\text{(≈ 220 mm}^2\text{)}.$$
Correct Answer: $70\pi\ \text{mm}^2$ (approximately $2.20\times10^{2}\ \text{mm}^2$)
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