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

NCERT Solution: Surface Areas and Volumes

If the lateral surface of a cylinder is 94.2 cm² and its height is 5 cm, then find (i) radius of its base (ii) its volume. (Use π = 3.14)

Given; curved surface area of cylinder = 94.2 sq cm, h = 5 cm
CSA of cylinder = 2 π rh
Or, 94.2 = 2 X (3.14) X r X 5
Or, r = 94.2/(2 X 3.14 X 5) = 3 cm

Now, volume of cylinder = π r² h
= 3.14 X 3² X 5 = 141.30 cubic cm

It costs Rs 2200 to paint the inner curved surface of a cylindrical vessel 10 m deep. If the cost of painting is at the rate of Rs 20 per m², find
(i) inner curved surface area of the vessel,
(ii) radius of the base,
(iii) capacity of the vessel.

Cost = Rs. 2200, rate = Rs. 20 per sq m, h = 10 m
Curved surface area = Cost/Rate
= 2200/20 = 110 sq cm

CSA of cylinder = 2 π rh
Or, 110 = 2 X (22/7) X r X 10
Or, r = (110 X 7)/(2 X 22 X 10) = 1.75 cm

Now, volume of cylinder = π r² h
= (22/7) X (1.75)² X 10 = 96.25 cubic m

The capacity of a closed cylindrical vessel of height 1 m is 15.4 litres. How many square metres of metal sheet would be needed to make it?

Capacity = 15.4 litre = 15400 cubic cm, h = 1 m = 100 cm
Volume of cylinder = π r² h
Or, 15400 = (22/7) X r² X 100
Or, r² = (15400 X 7)/(22 X 100) = 49
Or, r = 7 cm

Now, total surface area of cylinder = 2 π r(r + h)
= 2 X (22/7) X 7 (7 + 10)
= 2 X 22 X 17 = 748 sq cm

A lead pencil consists of a cylinder of wood with a solid cylinder of graphite filled in the interior. The diameter of the pencil is 7 mm and the diameter of the graphite is 1 mm. If the length of the pencil is 14 cm, find the volume of the wood and that of the graphite.

Radius of pencil = 3.5 mm, h = 14 cm = 140 mm
Volume of pencil = π r²h
= (22/7) X (3.5)² X 140 = 5390 cubic mm

Radius of lead = 0.5 mm, h = 140 mm
Volume of lead = π r² h
= (22/7) X (0.5)² X 140 = 110 cubic mm
Hence, volume of wood = 5390 – 110 = 5280 cubic mm

A patient in a hospital is given soup daily in a cylindrical bowl of diameter 7 cm. If the bowl is filled with soup to a height of 4 cm, how much soup the hospital has to prepare daily to serve 250 patients?

Given; r = 3.5 cm, h = 4 cm
Volume of cylinder = π r² h
= (22/7) X (3.5)² X 4 = 154 cubic cm
Volume of 250 bowls = 250 X 154 = 38500 cubic cm = 38.5 litre

Find the volume of the right circular cone with

(i) radius 6 cm, height 7 cm
(ii) radius 3.5 cm, height 12 cm

(i) radius 6 cm, height 7 cm

Given; r = 6 cm, h = 7 cm
Volume of cone = 1/3 π r²h
= (1/3) X (22/7) X 6² X 7 = 264 cubic cm

(ii) radius 3.5 cm, height 12 cm

Given; r = 3.5 cm, h = 12 cm
Volume of cone = 1/3 π r² h
= (1/3) X (22/7) X (3.5)² X 12 = 154 cubic cm

Find the capacity in litres of a conical vessel with

(i) radius 7 cm, slant height 25 cm
(ii) height 12 cm, slant height 13 cm

(i) radius 7 cm, slant height 25 cm

Answer: Given; r = 7 cm, l = 25 cm
Here; h² = l² - r²
= 25² - 7²
= 625 – 49 = 576
Or, h = 24 cm

Volume of cone = 1/3 π r² h
= (1/3) X (22/7) X 7² X 24 = 1232 cubic cm
= 1.232 litre

(ii) height 12 cm, slant height 13 cm

Answer: Given; h = 12 cm, l = 13 cm
Here, r² = l² - h²
= 13² - 12²
= 169 – 144 = 25
Or, r = 5 cm

Volume of cone = 1/3 π r² h
= (1/3) X (3.14) X 5² X 12 = 314 cubic cm
= 0.314 litre

The height of a cone is 15 cm. If its volume is 1570 cm³, find the radius of the base. (Use π = 3.14)

Given; volume = 1570 cubic cm, h = 15 cm, r = ?
Volume of cone = 1/3 π r²h
Or, 1570 = (1/3) X (3.14) X r² X 15
Or, r² = (1570 X 3)/(3.14 X 15) = 100
Or, r = 10 cm