Surface Areas and Volumes NCERT Solution, study material, CBSE Notes

NCERT Solution: Surface Areas and Volumes

The capacity of a cuboidal tank is 50000 litres of water. Find the breadth of the tank, if its length and depth are respectively 2.5 m and 10 m.

length = 2.5 m, depth = 10 m and volume = 50000 litres
1m³ = 1000 litres

∴ 50000 litres = 50000/1000 m³ = 50 m³
Breadth = Volume of cuboid/(Length x Depth)
= 50/(2.5 x 10) m
= 2 m

A village, having a population of 4000, requires 150 litres of water per head per day. It has a tank measuring 20m x 15m x 6m. For how many days will the water of this tank last?

Dimension of tank = 20m x 15m x 6m
l = 20 m , b = 15 m and h = 6 m
Capacity of the tank = lbh m³
= (20 x 15 x 6) m³
= 1800 m³
Water requirement per person per day =150 litres
Water required for 4000 person per day = (4000 x 150) l
                                                                = (4000 x 150) / 1000
                                                                = 600 m³
Number of days the water will last = Capacity of tank Total water required per day
                                                        =(1800/600) = 3
The water will last for 3 days.

A godown measures 40m x 25m x 15m. Find the maximum number of wooden crates each measuring 1.5m x 1.25m x 0.5m that can be stored in the godown.

Dimension of godown = 40 m x 25 m x 15 m
Volume of the godown = (40 x 25 x 15) m³ = 10000 m³
Dimension of crates = 1.5m x 1.25m x 0.5m
Volume of 1 crates = (1.5 x 1.25 x 0.5) m³ = 0.9375 m³
Number of crates that can be stored =Volume of the godown/Volume of 1 crate
= 10000/0.9375 = 10666.66 = 10666

A solid cube of side 12 cm is cut into eight cubes of equal volume. What will be the side of the new cube? Also, find the ratio between their surface areas.

Edge of the cube = 12 cm.
Volume of the cube = (edge)³ cm³
= (12 x 12 x 12) cm³
= 1728 cm³
Number of smaller cube = 8
Volume of the 1 smaller cube =1728/8 cm³ = 216 cm³
Side of the smaller cube = a
a³= 216
⇒ a = 6 cm
Surface area of the cube = 6 (side)²
Ratio of their surface area = (6 x 12 x 12)/(6 x 6 x 6)
= 4/1 = 4:1

A river 3 m deep and 40 m wide is flowing at the rate of 2 km per hour. How much water will fall into the sea in a minute?

Depth of river (h) = 3 m
Width of river (b) = 40 m
Rate of flow of water (l) = 2 km per hour = (2000/60) m per minute
= 100/3 m per minute
Volume of water flowing into the sea in a minute = lbh m³
= (100/3 x 40 x 3) m³
= 4000 m³

The circumference of the base of a cylindrical vessel is 132 cm and its height is 25 cm. How many litres of water can it hold? (1000 cm³ = 1l)

Given; circumference = 132 cm, h = 25 cm
Circumference = 2 π r
Or, 132 = 2 πr
Or, r = (132 X7)/(2 X 22) = 21 cm

Volume of cylinder = π r²h
= (22/7) X 21² X 25
= 34650 cubic cm = 34650/1000 litre
= 34.65 litre

The inner diameter of a cylindrical wooden pipe is 24 cm and its outer diameter is 28 cm. The length of the pipe is 35 cm. Find the mass of the pipe, if 1 cm³ of wood has a mass of 0.6 g.

Given; R = 28 cm, r = 24 cm, h = 35 cm
Volume of pipe = π R²h – π r²h
= π h (R² - r²)
= (22/7) X 35 (28² - 24²)
= 110 (28 + 24)(28 – 24)
= 110 X 52 X 4 = 22880 cubic cm
Mass = 22880 X 0.6 g = 13728 g

A soft drink is available in two packs – (i) a tin can with a rectangular base of length 5 cm and width 4 cm, having a height of 15 cm and (ii) a plastic cylinder with circular base of diameter 7 cm and height 10 cm. Which container has greater capacity and by how much?

Rectangular can: l = 5 cm, b = 4 cm, h = 15 cm
Volume of cubcoid = l X b X h
= 5 X 4 X 15 = 300 cubic cm

Cylindrical can: r = 35 cm, h = 10 cm
Volume of cylinder = π r² h
= (22/7) X 35² X 10
= 385 cubic cm
Difference = 385 – 300 = 85 cubic cm