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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.
Solution :
Given:
A cubical wooden block with edge length $l$. A hemispherical depression is cut out from one face of the cube such that the diameter of the hemisphere is equal to the edge of the cube ($d = l$).
To Find:
The total surface area of the remaining solid.
Step 1: Identify the components of the surface area.
The total surface area of the remaining solid consists of two parts:
1. The total surface area of the cube.
2. The curved surface area of the hemispherical depression (which is added to the total area).
3. The area of the circular top of the hemisphere (which must be subtracted from the cube's face because it is removed/hollowed out).
Step 2: Formulate the mathematical expressions.
Let the edge of the cube be $l$.
Total Surface Area of the cube = $6 \times (\text{edge})^2 = 6l^2$.
The diameter of the hemisphere is $l$, so the radius $r = \frac{l}{2}$.
Curved Surface Area (CSA) of the hemisphere = $2\pi r^2 = 2\pi \left(\frac{l}{2}\right)^2 = 2\pi \left(\frac{l^2}{4}\right) = \frac{\pi l^2}{2}$.
Area of the circular base of the hemisphere (to be subtracted) = $\pi r^2 = \pi \left(\frac{l}{2}\right)^2 = \frac{\pi l^2}{4}$.
Step 3: Calculate the total surface area of the remaining solid.
Total Surface Area = (Total Surface Area of Cube) - (Area of circular base) + (CSA of Hemisphere)
Total Surface Area = $6l^2 - \pi r^2 + 2\pi r^2$
Total Surface Area = $6l^2 + \pi r^2$
Substitute $r = \frac{l}{2}$ into the equation:
Total Surface Area = $6l^2 + \pi \left(\frac{l}{2}\right)^2$
Total Surface Area = $6l^2 + \frac{\pi l^2}{4}$
Step 4: Simplify the expression.
To add these terms, find a common denominator:
Total Surface Area = $\frac{24l^2}{4} + \frac{\pi l^2}{4}$
Total Surface Area = $\frac{l^2}{4} (24 + \pi)$
Final Answer: The total surface area of the remaining solid is $\frac{1}{4}l^2(\pi + 24)$ square units.
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.
- 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.
- 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
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