NCERT Class 10 Chapter 10 Surface Areas and Volumes Official CBSE Board Sample Problems Long Answer

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Question

Due to heavy floods in a state, thousands were rendered homeless. 50 schools collectively offered to the state government to provide place and the canvas for 1500 tents to be fixed by the government and decided to share the whole expenditure equally. The lower part of each tent is cylindrical of base radius 2.8 m and height 3.5m, with conical upper part of same base radius but of height 2.1 m.

If the canvas used to make the tents costs? 120 per sq. m, find the amount shared by each school to set up the tents. What value is generated by above problem?

Solution

Slant height of conical part

Area of tents = Curved surface area of cylindrical part + Curved surface area of conical part

Canvas required for 15(X) tents

Cost of 1500 tents

[.. Making tents costs? 120 per sq. m]

Share of each school

School authorities is concerned about safety of children and their families.

Question

A cylindrical tub, whose diameter is 12 cm and height 15 cm is full of ice-cream. The whole ice-cream is to be divided into 10 children in equal ice-cream cones, with conical base surmounted by hemispherical top. If the height of conical portion is twice the diameter of base, find the diameter of conical part of ice cream cone.

Solution

Volume of ice cream cylinder

Volume of 1 ice cream cone

[Height = 2 diameter)

Volume of ice cream 10 such cones

According to Quest ion,

Volume of 10 ice-cream cones = Volume of cylinder

Diameter of conical ice cream cup = 6 cm

Question

A hemispherical tank, of diameter 3 m, is full of water. It is being emptied by a pipe litre per second. How much time will it take to make the tank half empty?

Solution

Radius of hemispherical tank

Volume of water in hemispherical tank

Rate at water taken out in I second litre/second

Let time taken to empty to empty half the tank be Y see.

A. T. O.

Rate of flow of water t sec volume of water in the hemispherical tank

:. Time taken to empty half the tank =16 min. 30 sec.

Question

A tent consists of a frustum of a cone, surmounted by a cone. If the diameter of the upper and lower circular ends of the frustum be 14 m and 26 , respectively, the height of the frustum be 8 m and the slant height of the surmounted conical portion be 12 m. find the area of the canvas required to make the tent. (Assume that the radii of the upper circular end of the frustum and the base of the surmounted conical portion are equal.)

Solution

For the frustum

Upper diameter = 14 m

Upper Radius, r= 7m

Lower diameter = 26 m

Lower Radius, R= 13m

Height of the frustum h =8 m

Slant height I =

For the conical part

Radius of the base = r = 7 m

Slant height = L =12 m

Total surface area of the tent =curved area of frustum + Curved area of the cone

Question

A farmer connects a pipe of internal diameter 25 cm from a canal into a cylindrical tank in his field, which is 12 m in diameter and 2.5 m deep. If water flows through the pipe at the rate of 3.6 km/hr, then in how much time will the tank be filled?

Also, find the cost of water if the canal department charges at the rate of 0.07 per.

Solution

We have,

The radius of the cylindrical tank,

The depth of the tank,

The radius of the cylindrical pipe,

Speed of the flowing water,

Now,

Volume of water flowing out from the pipe in a hour

Also,

Volume of the tank

So, the time taken to fill the tank =

= 1.6h

Also, the cost of water

Question

A solid metallic sphere of diameter 21 cm is melted and recast into a number of smaller cones, each of diameter 3.5 cm and height 3 cm. Find the number of cones so formed.

Solution

Diameter of sphere = 21 cm

Radius of sphere

Volume of sphere=

Diameter of the cone

Radius of the cone

Height =3cm

72 72

Volume of each cone=

Total number of cones=

Question

The sum of the radius of the base and the height of a solid cylinder is 37 metres. If the total surface area of the cylinder be 1628 sq metres, find its volume.

Solution

Let r and h be the radius and height of the solid cylinder

Given

Now,

Total surface area of the cylinder

Volume of the cylinder

Question

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, ¡f8 cm of standing water is needed.

Solution

Speed = 10km/h

Water flow in 30 minutes = 10/2 = 5km = 5000m

Volume of water flow in 30 minutes cubic meter

Standing water depth

Irrigated area cubic meter hectares

Question

A well of diameter 4m and depth 21 m is dug. The earth taken out of it is evenly spread all around it in the shape of a circular ring of width 3m to form an embankment. Find the height of the embankment.

Solution

Volume of earth dug out =

Area of embankment =

Height of the embankment

Question

The height of a cone is 30 cm. A small cone is cut off at the top by a plane parallel to the base. If its volume be of the volume of the given cone, at what height above the base is the section made?

Solution

The height of a cone

The Height of a Cone

The height of a cone

Given height of the cone = 30 cm

Let the small cone is cut off at a height ‘h’ from the top

Let the radius of big cone be r1 and small cone be.

Volume of the big cone, 3

Volume of small cone,

Given

From the figure, [AA similarity criterion]

Hence equation (1), becomes

Thus at a height 20 cm above the base a small cone is cut.

Question

A vessel is in the form of an inverted cone. Its height is 8 cm and the radius is 5cm. 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 fifth of the water flows out. Find the number of lead shots dropped into the water.

Solution

Question

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?

Solution

Rate of flow of water

= 100 minutes

= 1 hour 40 minutes

Question

A conical vessel of radius 6cm and height 8cm is completely filled with water. A sphere is lowered into the water and its size is such that when it touches the sides, it is just immersed. What fraction of water overflows?

Solution

A conical vessel of radius

A Conical Vessel of Radius

A conical vessel of radius

Radius (R) of conical vessel = 6 cm

Height (H) of conical vessel = 8 cm

Volume of conical vessel (Vc)

Let the radius of the sphere be r cm.

In right APO’R, by Pythagoras theorem:

Hence

In right triangle MRO

Vofume of sphere (Vs)

Now,

Volume of the water = Volume of cone

Clearly, Volume of the water that flows out of cone is same as the volume of the sphere i.e. Vs.

Fraction of the water that flows out

Question

A metal bucket of height 24 cm is in the form of frustum of a cone radii of whose ends are 7 cm and 14 cm. Find the capacity of the bucket and the area of metal sheet required to make it.

Solution

(I). volume of frustum of cone

(ii). slant height of frustum I

Area of metallic sheet = CSA of frustum area of base

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