**Question 1.
**The height of a circular cylinder is 20 cm and the radius of its base is 7 cm. Find :

(i) the volume

(ii) the total surface area.

**Question 2.**

The inner radius of a pipe is 2.1 cm. How much water can 12 m of this pipe hold?

**Solution:**

**Question 3.**

**Solution:**

**Question 4.**

How many cubic meters of earth must be dug out to make a well 28 m deep and 2.8 m in diameter? Also, find the cost of plastering its inner surface at Rs 4.50 per sq meter.

**Solution:**

**Question 5.**

What length of solid cylinder 2 cm in diameter must be taken to recast into a hollow cylinder of external diameter 20 cm, 0.25 cm thick and 15 cm long?

**Solution:**

- Area Related to Circles Class 10 Formulas
- Surface Areas and Volumes Class 10 Formulas
- Constructions Class 10 Formulas
- Circles Class 10 Formulas
- Introduction to Trigonometry Class 10 Formulas
- Coordinate Geometry Class 10 Formulas
- Arithmetic Progression Class 10 Formulas
- Quadratic Equations Class 10 Formulas
- Pair Of Linear Equations in Two Variables Class 10 Formulas
- Polynomials Class 10 Formulas
- Real Numbers Class 10 Formulas
- Triangles Class 10 Formulas
- Some Applications of Trigonometry Class 10 Formulas
- Statistics Class 10 Formulas
- Probability Class 10 Formulas

**Question 6.**

A cylinder has a diameter of 20 cm. The area of curved surface is 100 sq cm. Find:

(i) the height of the cylinder correct to one decimal place.

(ii) the volume of the cylinder correct to one decimal place.

**Solution:**

**Question 7.**

A metal pipe has a bore (inner diameter) of 5 cm. The pipe is 5 mm thick all round. Find the weight, in kilogram, of 2 metres of the pipe if 1 cm3 of the metal weights 7.7 g.

**Solution:**

Inner radius of the pipe = r =\(\frac{5}{2}\) = 2.5 cm

External radius of the pipe = R = Inner radius of the pipe + Thickness of the pipe

= 2.5 cm + 0.5 cm

= 3 cm

Length of the pipe = h = 2 m= 200 cm

Volume of the pipe = External Volume – Internal Volume

Since 1cm^{3} of the metal weights 7.7 9,

∴ Weight of the pipe = (1728.6 × 7.7)g = \(\left(\frac{1728.6 \times 7.7}{1000}\right)\) kg = 13.31 kg

**Question 8.**

A cylindrical container with diameter of base 42 cm contains sufficient water to submerge a rectangular solid of iron with dimensions 22 cm x 14 cm 10.5 cm. Find the rise in level of the water when the solid is submerged.

**Solution:**

**Question 9.**

A cylindrical container with internal radius of its base 10 cm, contains water up to a height of 7 cm. Find the area of wetted surface of the cylinder.

**Solution:**

**Question 10.**

Find the total surface area of an open pipe of length 50 cm, external diameter 20 cm and internal diameter 6 cm.

**Solution:**

**Question 11.**

**Solution:**

**Question 12.
**The radius of a solid right circular cylinder increases by 20% and its height decreases by 20%. Find the percentage change in its volume.

**Question 13.
**The radius of a solid right circular cylinder decreases by 20% and its height increases by 10%. Find the percentage change in its :

(i) volume

(ii) curved surface area

**Question 14.
**Find the minimum length in cm and correct to nearest whole number of the thin metal sheet required to make a hollow and closed cylindrical box of diameter 20 cm and height 35 cm. Given that the width of the metal sheet is 1 m. Also, find the cost of the sheet at the rate of Rs. 56 per m.

Find the area of metal sheet required, if 10% of it is wasted in cutting, overlapping, etc.

**Question 15.
**3080 cm3 of water is required to fill a cylindrical vessel completely and 2310 cm

(i) radius of the vessel.

(ii) height of the vessel.

(iii) wetted surface area of the vessel when it is half-filled with water.

**Question 16.
**Find the volume of the largest cylinder formed when a rectangular piece of paper 44 cm by 33 cm is rolled along it :

(i) shorter side.

(ii) longer side.

**Question 17.
**A metal cube of side 11 cm is completely submerged in water contained in a cylindrical vessel with diameter 28 cm. Find the rise in the level of water.

**Question 18.
**A circular tank of diameter 2 m is dug and the earth removed is spread uniformly all around the tank to form an embankment 2 m in width and 1.6 m in height. Find the depth of the circular tank.

**Question 19.
**The sum of the inner and the outer curved surfaces of a hollow metallic cylinder is 1056 cm

**Question 20.
**The difference between the outer curved surface area and the inner curved surface area of a hollow cylinder is 352 cm

**Question 21.
**The sum of the heights and the radius of a solid cylinder is 35 cm and its total surface area is 3080 cm

**Question 22.
**The total surface area of a solid cylinder is 616 cm

**Question 23.
**A cylindrical vessel of height 24 cm and diameter 40 cm is full of water. Find the exact number of small cylindrical bottles, each of height 10 cm and diameter 8 cm, which can be filled with this water.

**Question 24.
**Two solid cylinders, one with diameter 60 cm and height 30 cm and the other with radius 30 cm and height 60 cm, are metled and recasted into a third solid cylinder of height 10 cm. Find the diameter of the cylinder formed.

**Question 25.**

**Solution:**

**Question 26.**

The given figure shows a solid formed of a solid cube of side 40cm and a solid cylinder of radius 20 cm and height 50 cm attached to the cube as shown.

Find the volume and the total surface area of the whole solid (Take π = 3.14)

**Solution:**

**Question 27.
**Two right circular solid cylinders have radii in the ratio 3 : 5 and heights in the ratio 2 : 3, Find the ratio between their :

(i) curved surface areas.

(ii) volumes.

**Question 28.
**A dosed cylindrical tank, made of thin ironsheet, has diameter = 8.4 m and height 5.4 m. How much metal sheet, to the nearest m

**Question 1.**

Find the volume of a cone whose slant height is 17 cm and radius of base is 8 cm.

**Solution:**

**Question 2.**

Solution:

**Question 3.**

The circumference of the base of a 12 m high conical tent is 66 m. Find the volume of the air contained in it.

**Solution:**

**Question 4.**

The radius and height of a right circular cone are in the ratio 5:12 and its volume is 2512 cubic cm. Find the radius and slant height of the cone. (Take π = 3.14)

**Solution:**

**Question 5.**

Two right circular cones x and y are made, x having three times the radius of y and y having half the volume of x. Calculate the ratio between the heights of x and y.

**Solution:**

**Question 6.**

The diameters of two cones are equal. If their slant heights are in the ratio 5:4, find the ratio of their curved surface areas.

**Solution:**

**Question 7.**

There are two cones. The curved surface area of one is twice that of the other. The slant height of the latter is twice that of the former. Find the ratio of their radii.

**Solution:**

**Question 8.**

A heap of wheat is in the form of a cone of diameter 16.8 m and height 3.5 m. Find its volume. How much cloth is required to just cover the heap?

**Solution:**

**Question 9.**

Find what length of canvas, 1.5 m in width, is required to make a conical tent 48 m in diameter and 7 m in height. Given that 10% of the canvas is used in folds and stitching. Also, find the cost of the canvas at the rate of Rs. 24 per meter.

**Solution:**

**Question 10.**

A solid cone of height 8 cm and base radius 6 cm is melted and re-casted into identical cones, each of height 2 cm and diameter 1 cm. Find the number of cones formed.

**Solution:**

**Question 11.**

**Solution:**

**Question 12.**

**Solution:**

**Question 13.**

A vessel, in the form of an inverted cone, is filled with water to the brim. Its height is 32 cm and diameter of the base is 25.2 cm. Six equal solid cones are dropped in it, so that they are fully submerged. As a result, one-fourth of water in the original cone overflows. What is the volume of each of the solid cones submerged?

**Solution:**

**Question 14.**

**Solution:**

**Question 1.**

**Solution:**

**Question 2.**

**Solution:**

**Question 3.**

A spherical ball of lead has been melted and made into identical smaller balls with radius equal to half the radius of the original one. How many such balls can be made?

**Solution:**

**Question 4.**

How many balls each of radius 1 cm can be made by melting a bigger ball whose diameter is 8 cm.

**Solution:**

**Question 5.**

8 metallic sphere; each of radius 2 mm, are melted and cast into a single sphere. Calculate the radius of the new sphere.

**Solution:**

**Question 6.**

The volume of one sphere is 27 times that of another sphere. Calculate the ratio of their:

(i) radii

(ii) surface areas

**Solution:**

**Question 7.**

If the number of square centimeters on the surface of a sphere is equal to the number of cubic centimeters in the volume, what is the diameter of the sphere?

**Solution:**

**Question 8.**

**Solution:**

**Question 9.**

The internal and external diameters of a hollow hemi-spherical vessel are 21 cm and 28 cm respectively. Find:

(i) internal curved surface area

(ii) external curved surface area

(iii) total surface area

(iv) volume of material of the vessel.

**Solution:**

**Question 10.**

A solid sphere and a solid hemi-sphere have the same total surface area. Find the ratio between their volumes.

**Solution:**

**Question 11.**

Metallic spheres of radii 6 cm, 8 cm and 10 cm respectively are melted and recasted into a single solid sphere. Taking ∏ = 3.1, find the surface area of solid sphere formed.

**Solution:**

**Question 12.**

The surface area of a solid sphere is increased by 21% without changing its shape. Find the percentage increase in its:

(i) radius

(ii) volume

**Solution:**

**Question 1.
**A solid sphere of radius 15 cm is melted and recast into solid right circular cones of radius 2.5 cm and height 8 cm. Calculate the number of cones recast.

**Question 2.**

A hollow sphere of internal and external diameters 4 cm and 8 cm respectively is melted into a cone of base diameter 8 cm. Find the height of the cone.

**Solution:**

**Question 3.**

The radii of the internal and external surfaces of a metallic spherical shell are 3 cm and 5 cm respectively. It is melted and recast into a solid right circular cone of height 32 cm. find the diameter of the base of the cone.

**Solution:**

**Question 4.**

Total volume of three identical cones is the same as that of a bigger cone whose height is 9 cm and diameter 40 cm. find the radius of the base of each smaller cone, if height of each is 108 cm.

**Solution:**

**Question 5.**

A solid rectangular block of metal 49 cm by 44 cm by 18 cm is melted and formed into a solid sphere. Calculate the radius of the sphere.

**Solution:**

**Question 6.**

A hemi-spherical bowl of internal radius 9 cm is full of liquid. This liquid is to be filled into conical shaped small containers each of diameter 3 cm and height 4 cm. How many containers are necessary to empty the bowl?

**Solution:**

Radius of hemispherical bowl = 9 cm

Volume = \(\frac{1}{2} \times \frac{4}{3} \pi r^{3}=\frac{2}{3} \pi 9^{3}=\frac{2}{3} \pi \times 729=486 \pi \mathrm{cm}^{2}\)

Diameter each of cylindrical bottle = 3 cm

Radius = \(\frac{3}{2}\)cm, and height = 4 cm

∴ Volume of bottle = \(\frac{1}{3} \pi \pi^{2} n=\frac{1}{3} \pi \times\left(\frac{3}{2}\right)^{2} \times 4=3 \pi\)

∴ No. of bottles = \(\frac{486 \pi}{3 \pi}=162\)

**Question 7.**

A hemispherical bowl of diameter 7.2 cm is filled completely with chocolate sauce. This sauce is poured into an inverted cone of radius 4.8 cm. Find the height of the cone if it is completely filled.

**Solution:**

**Question 8.**

A solid cone of radius 5 cm and height 8 cm is melted and made into small spheres of radius 0.5 cm. Find the number of spheres formed.

**Solution:**

**Question 9.**

**Solution:**

**Question 10.**

A solid metallic cone, with radius 6 cm and height 10 cm, is made of some heavy metal A. In order to reduce weight, a conical hole is made in the cone as shown and it is completely filled with a lighter metal B. The conical hole has a diameter of 6 cm and depth 4 cm. Calculate the ratio of the volume of the metal A to the volume of metal B in the solid.

**Solution:**

**Question 11.**

A hollow sphere of internal and external radii 6 cm and 8 cm respectively is melted and recast into small cones of base radius 3 cm and height 8 cm. Find the number of cones.

**Solution:**

**Question 12.
**The surface area of a solid metallic sphere is 2464 cm

(i) the radius of the sphere.

(ii) the number of cones recast. (Take π = \(\frac{22}{7}\))

**Question 1.**

A cone of height 15 cm and diameter 7 cm is mounted on a hemisphere of same diameter. Determine the volume of the solid thus formed.

**Solution:**

**Question 2.**

A buoy is made in the form of a hemisphere surmounted by a right circular cone whose circular base coincides with the plane surface of the hemisphere. The radius of the base of the cone is 3.5 m and its volume is two-third the volume of hemisphere. Calculate the height of the cone and the surface area of the buoy, correct to two decimal places.

**Solution:**

**Question 3.**

**Solution:**

**Question 4.**

The cubical block of side 7 cm is surmounted by a hemisphere of the largest size. Find the surface area of the resulting solid.

**Solution:**

**Question 5.**

A vessel is in the form of an inverted cone. Its height is 8 cm and the radius of the top which is open is 5 cm. It is filled with water. 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.

**Solution:**

**Question 6.**

A hemispherical bowl has negligible thickness and the length of its circumference is 198 cm. find the capacity of the bowl.

**Solution:**

**Question 7.**

Find the maximum volume of a cone that can be carved out of a solid hemisphere of radius r cm.

**Solution:**

**Question 8.**

**Solution:**

**Question 9.**

A solid hemisphere of diameter 28 cm is melted and recast into a number of identical solid cones, each of diameter 14 cm and height 8 cm. Find the number of cones so formed.

**Solution:**

**Question 10.**

A cone and a hemisphere have the same base and same height. Find the ratio between their volumes.

**Solution:**

**Question 1.**

From a solid right circular cylinder with height 10 cm and radius of the base 6 cm, a right circular cone of the same height and same base are removed. Find the volume of the remaining solid.

**Solution:**

**Question 2.**

From a solid cylinder whose height is 16 cm and radius is 12 cm, a conical cavity of height 8 cm and of base radius 6 cm is hollowed out. Find the volume and total surface area of the remaining solid.

**Solution:**

**Question 3.**

A circus tent is cylindrical to a height of 4 m and conical above it. If its diameter is 105 m and its slant height is 80 m, calculate the total area of canvas required. Also, find the total cost of canvas used at Rs 15 per meter if the width is 1.5 m

**Solution:**

**Question 4.**

A circus tent is cylindrical to a height of 8 m surmounted by a conical part. If total height of the tent is 13 m and the diameter of its base is 24 m; calculate:

(i) total surface area of the tent

(ii) area of canvas, required to make this tent allowing 10% of the canvas used for folds and stitching.

**Solution:**

Height of the cylindrical part = H = 8 m

Height of the conical part = h = (13 – 8)m = 5 m

Diameter = 24 m → radius = r = 12 m

Slant height of the cone = l

Slant height of cone = 13 m

(i) Total surface area of the tent = 2πrh + πrl = πr(2h + l)

(ii)Area of canvas used in stitching = total area

**Question 5.**

A cylindrical boiler, 2 m high, is 3.5 m in diameter. It has a hemispherical lid. Find the volume of its interior, including the part covered by the lid.

**Solution:**

**Question 6.**

**Solution:**

**Question 7.**

**Solution:**

**Question 8.**

A cylindrical container with diameter of base 42 cm contains sufficient water to submerge a rectangular solid of iron with dimensions 22 cm x 14 cm x 10.5 cm. Find the rise in level of the water when the solid is submerged.

**Solution:**

**Question 9.**

Spherical marbles of diameter 1.4 cm are dropped into beaker containing some water and are fully submerged. The diameter of the beaker is 7 cm. Find how many marbles have been dropped in it if the water rises by 5.6 cm.

**Solution:**

**Question 10.**

**Solution:**

**Question 11.**

**Solution:**

**Question 12.**

**Solution:**

**Question 13.**

**Solution:**

**Question 14.**

A cylindrical can, whose base is horizontal and of radius 3.5 cm, contains sufficient water so that when a sphere is placed in the can, the water just covers the sphere. Given that the sphere just fits into the can, calculate:

(i) the total surface area of the can in contact with water when the sphere is in it;

(ii) the depth of water in the can before the sphere was put into the can.

**Solution:**

**Question 15.**

A hollow cylinder has solid hemisphere inward at one end and on the other end it is closed with a flat circular plate. The height of water is 10 cm when flat circular surface is downward. Find the level of water, when it is inverted upside down, common diameter is 7 cm and height of the cylinder is 20 cm.

**Solution:**

**Question 1.**

What is the least number of solid metallic spheres, each of 6 cm diameter, that should be melted and recast to form a solid metal cone whose height is 45 cm and diameter is 12 cm?

**Solution:**

**Question 2.**

A largest sphere is to be carved out of a right circular cylinder of radius 7 cm and height 14 cm. Find the volume of the sphere. (Answer correct to the nearest integer)

**Solution:**

**Question 3.**

A right circular cylinder having diameter 12 cm and height 15 cm is full of ice-cream. The ice-cream is to be filled in identical cones of height 12 cm and diameter 6 cm having a semi-spherical shape on the top. Find the number of cones required.

**Solution:**

**Question 4.**

A solid is in the form of a cone standing on a hemisphere with both their radii being equal to 8 cm and the height of cone is equal to its radius. Find in terms of π, the volume of the solid.

**Solution:**

**Question 5.**

The diameter of a sphere is 6 cm. It is melted and drawn into a wire of diameter 0.2 cm. Find the length of wire.

**Solution:**

**Question 6.**

Determine the ratio of the volume of a cube to that of a sphere which will exactly fit inside the cube.

**Solution:**

**Question 7.**

**Solution:**

**Question 8.**

**Solution:**

**Question 9.**

**Solution:**

**Question 10.**

A cylindrical water tank of diameter 2.8m and height 4.2m is being fed by a pipe of diameter 7 cm through which water flows at the rate of 4m/s. Calculate, in minutes, the time it takes to fill the tank.

**Solution:**

**Question 11.**

**Solution:**

**Question 12.**

**Solution:**

**Question 13.**

A solid, consisting of a right circular cone, standing on a hemisphere, is placed upright, in a right circular cylinder, full of water, and touches the bottom. Find the volume of water left in the cylinder, having given that the radius of the cylinder is 3 cm and its height is 6 cm; the radius of the hemisphere is 2 cm and the height of the cone is 4 cm. Give your answer to the nearest cubic centimeter.

**Solution:**

**Question 14.**

**Solution:**

**Question 15.**

**Solution:**

**Question 16.**

**Solution:**

**Question 17.**

**Solution:**

**Question 18.
**Two solid spheres of radii 2 cm and 4 cm are melted and recast into a cone of height 8 cm. Find the radius of the cone so formed.

**Question 19.
**A certain number of metallic cones, each of radius 2 cm and height 3 cm, are melted and recast into a solid sphere of radius 6 cm. Find the number of cones used.

Let the number of cones melted be n.

Let the radius of sphere be r

Radius of cone be r

And, height of the cone be h = 3 cm

Volume of sphere = n (Volume of a metallic cone)

**Question 20.
**A conical tent is to accommodate 77 persons. Each person must have 16m