**1. 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.**

**Ans.**Diameter of the hollow hemisphere = 14 cm

Radius of the hollow hemisphere = = 7 cm

Total height of the vessel = 13 cm

Height of the hollow cylinder = 13 – 7 = 6 cm

Inner surface area of the vessel

= Inner surface area of the hollow hemisphere + Inner surface area of the hollow cylinder

= =

= = 26 x 22 = 572 cm

^{2}

**2. 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.**

**Ans.** Radius of the cone = 3.5 cm

Radius of the hemisphere = 3.5 cm

Total height of the toy = 15.5 cm

Height of the cone = 15.5 – 3.5 = 12 cm

Slant height of the cone =

=

= = 12.5 cm

TSA of the toy = CSA of hemisphere + CSA of cone

= =

= = = = 214.5 cm^{2}

**3. 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.**

**Ans.** Greatest diameter of the hemisphere = Side of the cubical block = 7 cm

TSA of the solid = External surface area of the cubical block + CSA of hemisphere

=

=

= =

= = 294 + 38.5 = 332.5 cm^{2}

**4. A hemispherical depression is cut out from one face of a cubical wooden block such that the diameter of the hemisphere is equal to the edge of the cube. Determine the surface area of the remaining solid**

**Ans.** Diameter of the hemisphere = , therefore radius of the hemisphere =

Also, length of the edge of the cube =

Surface area of the remaining solid =

= = =

**5. Rachel, an engineering student, was asked to make a model shaped like a cylinder with two cones attached at its two ends by using a thin aluminium sheet. The diameter of the model is 3 cm and its length is 12 cm. If each cone has a height of 2 cm, find the volume of air contained in the model that Rachel made. (Assume the outer and inner dimensions of the model to be nearly the same.)**

**Ans.** For upper conical portion, Radius of the base = 1.5 cm

Height = 2 cm

Volume = = = cm^{3}

For lower conical portion, Volume = cm^{3}

For central cylindrical portion

Radius of the base = 1.5 cm

Height = 12 – (2 + 2) = 8 cm

Volume = = = cm^{3}

Volume of the model = + +

=

= = 66 cm^{3}

**6. A gulabjamun, contains sugar syrup up to about 30% of its volume. Find approximately how much syrup would be found in 45 gulab jamuns, each shaped like a cylinder with two hemispherical ends, with length 5 cm and diameter 2.8 cm (see figure).**

**Ans.**Volume of a gulabjamun =

=

=

=

= = cm

^{3}

Volume of 45 gulabjamuns =

=

= 1127.28 cm

^{3}

Volume of syrup =

= 338.184 cm

^{3}= 338 cm

^{3}(approx.)

**7. A vessel is in the form of inverted cone. Its height is 8 cm and the radius of the top, which is open, is 5 cm. It is filled with water up to the brim. When lead shots, each of which is a sphere of radius 0.5 cm are dropped into the vessel, one-fourth of the water flows out. Find the number of lead shots dropped in the vessel.**

**Ans. For cone**, Radius of the top = 5 cm and height = 8 cm

Volume of cone = =

= cm^{3}

**For spherical lead shot**, Radius (R) = 0.5 cm

Volume of spherical lead shot = =

= cm^{3}

Volume of water that flows out = Volume of the cone

= = cm^{3}

Let the number of lead shots dropped in the vessel be

= 100

**8. A solid iron pole consists of a cylinder of height 220 cm and base diameter 24 cm, which is surmounted by another cylinder of height 60 cm and radius 8 cm. Find the mass of the pole, given that 1 cm ^{3} of iron has approximately 8 g mass. **

**Ans. For lower cylinder**, Base radius = = 12 cm

And Height = 220 cm

Volume = = = cm

^{3}

**For upper cylinder**, Base Radius (R) = 8 cm

And Height (H) = 60 cm

Volume = = = cm

^{3}

Volume of the solid Iron pole = V of lower cylinder + V of upper cylinder

= + =

= 35520 x 3.14 = 111532.8 cm

^{3}

**9. A solid consisting of a right circular cone of height 120 cm and radius 60 cm standing on a hemisphere of radius 60 cm is placed upright in a right circular cylinder full of water such that it touches the bottom. Find the volume of water left in the cylinder, if the radius of the cylinder is 60 cm and its height is 180 cm.**

**Ans.** **For right circular cone**, Radius of the base = 60 cm

And Height = 120 cm

Volume = = = cm^{3}

**For Hemisphere**, Radius of the base = 60 cm

Volume = = = cm^{3}

**For right circular cylinder**, Radius of the base = 60 cm

And Height = 180 cm

Volume = = = cm^{3}

Now, V of water left in the cylinder

= V of right circular cylinder – (V of right circular cone + V of hemisphere)

= – ( + )

= cm^{3}

= m^{3}

= = 1.131 m^{3} (approx.)

**10. A spherical glass vessel has a cylindrical neck 8 cm long, 2 cm in diameter; the diameter of the spherical part is 8.5 cm. By measuring the amount of water it holds, a child finds its volume to be 345 cm ^{3}. Check whether she is correct, taking the above as the inside measurements and = 3.14.**

**Ans.**Amount of water it holds =

=

=

= 321.39 + 25.12

= 346.51 cm

^{3}

Hence, she is not correct. The correct volume is 346.51 cm

^{3}.

**11. A cylindrical bucket, 32 cm and high and with radius of base 18 cm, is filled with sand. This bucket is emptied on the ground and a conical heap of sand is formed. If the height of the conical heap is 24 cm, find the radius and slant height of the heap.**

**Ans. For cylindrical bucket**, Radius of the base = 18 cm and height = 32 cm

Volume = = = cm^{3}

**For conical heap**, Height = 24 cm

Let the radius be cm.

Then, Volume = = = cm^{3}

According to question, Volume of bucket = Volume of conical heap

=

= 1296

= 36 cm

Now, Slant height =

= =

= = cm

**12. Water in a canal 6 m wide and 1.5 m deep is flowing with a speed of 10 km/h. How much area will it irrigate in 30 minutes, if 8 cm of standing water is needed?**

**Ans.** **For canal**, Width = 6 m and Depth = 1.5 m = m

Speed of flow of water = 10 km/h = 10 x 1000 m/h = 10000 m/h

= m/min = m/min

Speed of flow of water in 30 minutes = m/min

Volume of water that flows in 30 minutes = = 45000 m^{3}

The area it will irrigate = = 562500 m^{2}

= hectares = 56.25 hectares

**13. A farmer connects a pipe of internal diameter 20 cm from a canal into a cylindrical tank in her field, which is 10 m in diameter and 2 m deep. If water flows through the pipe at the rate of 3 km/h, in how much time will the tank be filled?**

**Ans.** For cylindrical tank, Diameter = 10 m

Radius = 5 m and Depth = 2 m

Volume = = = m^{3}

Rate of flow of water = 3 km/h = 3000 m/h

= m/min = 50 m/min

For pipe, Internal diameter = 20 cm, therefore radius= 10 cm = 0.1 m

Volume of water that flows per minute =

= = m^{3}

Required time = = 100 minutes

**14. A metallic right circular cone 20 cm high and whose vertical angle is is cut into two parts at the middle of its height by a plane parallel to its base. If the frustum so obtained be drawn into a wire of diameter cm, find the length of the wire.**

**Ans.**

cm

cm

= 10 cm

Volume =

=

=

= = cm^{3}

Diameter of the wire = cm

Radius of the wire = cm

Let the length of the wire be cm.

Then, Volume of the wire = =

= cm^{3}

According to the question,

=

796444.44 cm = 7964.4 m

**15. The diameter of metallic sphere in 6 cm. The sphere in melted and drawn into a wire of uniform cross section. If the length of wire is 36 cm, find its radius.**

**Ans.** Diameter of sphere = 6 cm

Volume

Let r_{1} be radius of wire

Volume of wire =

**16. Water flows at the rate of 10 metre per minute through a cylindrical pipe having its diameter at 5mm. How much time will it take to fill a conical vessel where diameter of base is 40 cm and depth 24 cm?**

**Ans.** We have volume of the water that flows out in one minute

= Volume of cylinder of diameter 5 mm and length 10 m.

Volume of cylinder =

Volume of conical vessel

Suppose the conical vessel is filled in x minutes

Volume of the water flows out in x minutes

= Volume of conical vessel

Minutes

Minutes 12 seconds

**17. The radius of the base and the height of solid right cylinder are in the ratio 2:3 and its volume is 1617 cu.cm. Find the total surface area of the cylinder. **

**Ans.** Let r be the radius of the base and h be the height of the solid right circular cylinder.

Volume of the cylinder =

Surface area of cylinder =

**18. A toy is in form of a cone mounted on a hemisphere of common base radius 7 cm. The total height of the toy is 31 cm, find the total surface area of the toy.**

**Ans.** Total surface area of toy = C.A.S. of hemisphere + C.S.A. of cone

=

Here,

T.S.A. of toy =

**19. A well 3.5 m in diameter and 20 m deep be dug in rectangular field 20 m by 14 m. The earth taken out is spread evenly on the field. Find the level of the earth raised in the field.**

**Ans.** Radius of well

Depth of well

Volume of earth taken out

Area of field

Area of field excluding well

Level of earth raised

**20. A solid sphere of radius 6 cm is melted into a hollow cylinder of uniform thickness. If the external radius of the base of cylinder is 5 cm and its height is 32 cm, find the uniform thickness of the cylinder. Ans.** Volume of sphere

Let the internal radius of cylinder

External radius

Volume

Volume of the hollow cylinder = Volume of sphere

**21. A medicine capsule is in the shape of a cylinder with two hemispheres stuck it each of its ends. The length of the entire capsule is 14 mm and the diameter of the capsule is 5 mm. Find its surface area. Ans.** Uniform thickness of cylinder = 5 – 4=1 cm

The length of capsule = 14 mm

Length of cylindrical portion of capsule= 9 mm

Total surface area =

**22. A pen stand made of wood is in the shape of a cuboid with four conical depression to hold pens. The dimensions of the cuboid are 15 cm by 10 cm by 3.5 cm. The radius of each of the dimensions is 0.5 cm and the depth is 1.4 cm. Find the volume of the wood in the entire stand.**

**Ans.** Required volume = volume of cuboids – 4 [V. of one depression]

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

**Ans.** The length of capsule = 14 mm

Length of cylindrical portion of capsule= 9 mm

Total surface area =

**24. A spherical glass vessel has a cylindrical neck 8 cm long, 2 m in diameter, the diameter of the spherical part 8.5 cm. by measuring the amount of water it holds, a child finds its volume to be 345 cm ^{3} check whether she is correct, taking the above as side measurements.**

**Ans.**For cylindrical part

For spherical part:

Radius (R)

Volume of glass solid = Volume of cylindrical part + Volume of the spherical part

**25. Metallic sphere of radii 6 cm, 8 cm and 10 cm respectively are melted to form a single solid sphere. Find the radius of the resulting sphere.**

**Ans.** Sum of the volumes of three given spheres =

Let R be the radius of single solid sphere, since volume remains the same

**26. A shuttle cock used for playing badminton has the shape of a frustum of a cone mounted on a hemisphere. The external diameter of the frustum are 5 cm and 2 cm. The height of the entire shuttle cock is 7 cm. Find the external surface area.**

**Ans.** External surface area= C.S.A. of frustum of the come + S.A. of hemisphere

=

**27. How many silver coins 1.75 cm in diameter and of thickness 2 mm must be melted to form a cuboid 5.5 cm10 cm3.5 cm?**

**Ans. **Volume of the cuboid

Radius of the coin

Thickness

Volume of one coin

Required number of coins =

s

**28. A container like a right circular having diameter 12cm and height 15cm is full of ice-cream. The ice-cream is to be filled in cones of height 12cm and diameter 6cm having a hemispherical shape on the top. Find number of such cones which can be filled with ice-cream. Ans.** Volume of cylinder

Diameter of cone = 12 cm

Volume of ice cream = Volume of ice-cream cone + Volume of hemispherical top of ice-cream

Number of ice-Cream cones

ice-cream cones

**29. Water flowing at the rate of 15 km per hour through a pipe of diameter 14cm into a rectangular tank which is 50m long and 44m wide. Find the time in which the level of water in the tank will rise by 21cm.**

**Ans.** 1 Km = 1000 m

Volume of cylinder =

Radius

Volume of water flowing through the cylindrical pipe in an hour at the rate of 15km/hr

Volume of cuboid =

Volume of required quantity of water in the tank

Since of water falls into tank in 1 hour

of water falls into tank in hours

**30. A solid cylinder of diameter 12cm and height 15cm is melted and recast into toys with the shape of a right circular cone mounted on a hemisphere of radius 3cm. If the height of the boy is 12cm, find the number of toys so formed. Ans. **