Chapter 13. Surface Areas and Volumes Exercise 13.1 solved solution

**Q1. 2 cubes each of volume 64 cm3 are joined end to end. Find the surface area of the resulting cuboid.**

**Solution:**

The volume of cube = 64 cm^{3}

Joining two surfaces;

l = 4 + 4 = 8 cm

b = 4 cm

h = 4 cm

The Surface Area of such formed cuboid = 2(lb + bh + lh)

= 2(8×4 + 4×4 + 8×4)

= 2(32 + 16 + 32)

= 2×80

= 160 cm^{2}

Therefore, The area obtained from this cuboid is 160 cm^{2}

**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.**

**Solution: **

**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.**

**Solution:**

The radius of hemisphererr = 3.5 cm

The radius of conical partr = 3.5 cm

The height of conical partrh = 15.5 – 3.5 = 12 cm

**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.**

**Solution**:

The edge of cubical block = 7 cm

The maximum diameter of the blockd = 7 cm

**The total surface area of block = The Area of cuboidal block + The Area of hemisphere – Area of a circle covered with hemisphere **

= 294 + 38.5 = 332.5 cm^{2}

Therefore, total surface area of block = 332.5 cm^{2}

**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**:

Let the diameter of hemisphered = *l* unit

And the edge of cubea = *l* unit

(Hence, diameter of hemisphere and the edge of cube are same)

Total surface Area of remaining solid = Area of cuboidal block + Area of hemisphere – Area of a circle covered with hemisphere

**Q6. A medicine capsule is in the shape of a cylinder with two hemispheres stuck to each of its ends (see Fig. 13.10). The length of the entire capsule is 14 mm and the diameter of the capsule is 5 mm. Find its surface area.**

**Solution:**

The height of cylinder = length of capsule – 2(2.5 mm)

= 14 – 5 [Hence diameter = 2.5 mm]

= 9 mm

Total Surface area of capsule = 2(curve surface area of hemisphere) + curve surface area of cylinder

**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 Rs 500 per m2. (Note that the base of the tent will not be covered with canvas.)**

**Solution:**

** **

The diameter of the cylindrical part of tent = 4 m

So, radius = 2 m

The height of cylindrical part = 2.1 m

The slant height of cone = 2.8 m

And radius = 2 m

The area of required canvas = the curved surface area of cylindrical part + curved surface area of conical part

= 2πrh + πrl

**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}.**

**Solution:**

The height of cylinder h = 2.4 cmcm

The diameter of cylinder = 1.4 cm

So, radius of cylinder = 0.7 cm

The height of hollowed out = 2.4 cm

Radius = 0.7 cm

**Total surface area of remaining solid **= Curved surface area of cylinder + curved surface area of cone + Area of the bottom of solid

= 2πrh + πrl + πr^{2}

= πr(2h + l + r)

The total surface area of the remaining solid to the nearest cm^{2} = 18 cm^{2 }**Answer**

**Q9. A wooden article was made by scooping out a hemisphere from each end of a solid cylinder, as shown in Fig. 13.11. 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.**

**Solution: **

The height of cylinder = 10 cm

The radius (r) of base = 3.5 cm

The radius (r) of hemisphere = 3.5 cm

Total surface area = Curved surface area of cylinder + Curved surface area of top of hemisphere + Curved surface area of bottom of hemisphere

= 2πrh + 2πr^{2} + 2πr^{2}

= 2πr(h h + rr + rr )

= 2πr(hh + 2rr)

**Total surface area is 374 cm ^{2}**