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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.
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.
Solution :
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$^2$.
More Questions from Class 10 Mathematics Surface Areas and Volumes EXERCISE 12.1
- Q1: 2 cubes each of volume $64$ cm$^3$ are joined end to end. Find the surface area of the resulting cuboid.
- Q2: A vessel is in the form of a hollow hemisphere mounted by a hollow cylinder. The diameter of the hemisphere is $14$ cm and the total height of the vessel is $13$ cm. Find the inner surface area of the vessel.
- Q3: A toy is in the form of a cone of radius $3.5$ cm mounted on a hemisphere of same radius. The total height of the toy is $15.5$ cm. Find the total surface area of the toy.
- Q4: A cubical block of side $7$ cm is surmounted by a hemisphere. What is the greatest diameter the hemisphere can have? Find the surface area of the solid.
- Q5: 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.
- Q7: 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 m$^2$. (Note that the base of the tent will not be covered with canvas.)
- Q8: 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 cm$^2$.
- Q9: 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.
CBSE Solutions for Class 10 Mathematics Surface Areas and Volumes
Chapters in CBSE - Class 10 Mathematics
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