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NCERT Solutions for Class 10 Maths Chapter 12 - Surface Areas and Volumes (Exercise 12.5)

Page No. 258

Q 1. A copper wire, 3 mm in diameter, is wound about a cylinder whose length is 12 cm, and diameter 10 cm, so as to cover the curved surface of the cylinder. Find the length and mass of the wire, assuming the density of copper to be 8.88 g per cm3.
Sol. Since, diameter of the cylinder = 10 cm
∴ Radius of the cylinder (r) =  10/2 cm = 5 cm
⇒ Length of wire in completely one round
2πr = 2 × 3.14 × 5 cm = 31.4 cm
∵ Diameter of wire = 3 mm = 3/10 cm
∴ The thickness of cylinder covered in one round = 3/10 cm
⇒ Number of rounds (turns) of the wire to cover 12 cm
NCERT Solutions for Class 10 Maths Chapter 12 - Surface Areas and Volumes (Exercise 12.5)
∴ Length of wire required to cover the whole surface = Length of wire required to complete 40 rounds
= 40 × 31.4 cm = 1256 cm
Now, radius of the wire  NCERT Solutions for Class 10 Maths Chapter 12 - Surface Areas and Volumes (Exercise 12.5)
∴ Volume of wire = πr2l
NCERT Solutions for Class 10 Maths Chapter 12 - Surface Areas and Volumes (Exercise 12.5)
∵ Density of wire = 8.88 gm/cm3
∴Weight of the wire = [Volume of the wire] × Density
NCERT Solutions for Class 10 Maths Chapter 12 - Surface Areas and Volumes (Exercise 12.5)
= 788 g (approx.)

Q 2. A right triangle, whose sides are 3 cm and 4 cm (other than hypotenuse) is made to revolve about its hypotenuse. Find the volume and surface area of the double cone so formed. Choose the value of π as found appropriate.
NCERT Solutions for Class 10 Maths Chapter 12 - Surface Areas and Volumes (Exercise 12.5)
Sol. Let us consider the rt ΔBAC, rt. angled at A such that
AB = 3 cm, AC = 4 cm
∴ Hypotenuse NCERT Solutions for Class 10 Maths Chapter 12 - Surface Areas and Volumes (Exercise 12.5)
Obviously, we have obtained two cones on the same base AA′ such that the radius = DA or DA'
NCERT Solutions for Class 10 Maths Chapter 12 - Surface Areas and Volumes (Exercise 12.5)NCERT Solutions for Class 10 Maths Chapter 12 - Surface Areas and Volumes (Exercise 12.5)
NCERT Solutions for Class 10 Maths Chapter 12 - Surface Areas and Volumes (Exercise 12.5)
NCERT Solutions for Class 10 Maths Chapter 12 - Surface Areas and Volumes (Exercise 12.5)
Now, volume of the double cone
NCERT Solutions for Class 10 Maths Chapter 12 - Surface Areas and Volumes (Exercise 12.5)
Surface area of the double cone
NCERT Solutions for Class 10 Maths Chapter 12 - Surface Areas and Volumes (Exercise 12.5)

Q 3. A cistern, internally measuring 150 cm × 120 cm × 110 cm, has 129600 cm3 of water in it. Porous bricks are placed in the water until the cistern is full to the brim. Each brick absorbs one-seventeenth of its own volume of water. How many bricks can be put in without overflowing the water, each being 22.5 cm × 7.5 cm × 6.5 cm?
Sol. ∵ Dimensions of the cistern are 150 cm, 120 cm and 110 cm.
∴ Volume of the cistern = 150 × 120 × 110 cm3 = 1980000 cm3
Volume of water contained in the cistern = 129600 cm3
∴ Free space (volume) which is not filled with water = 1980000 − 129600 cm3 = 1850400 cm3
Now, Volume of one brick = 22.5 × 7.5 × 6.5 cm3
NCERT Solutions for Class 10 Maths Chapter 12 - Surface Areas and Volumes (Exercise 12.5)
∴ Volume of water absorbed by one brick  NCERT Solutions for Class 10 Maths Chapter 12 - Surface Areas and Volumes (Exercise 12.5)
Let ‘n’ bricks can be put in the cistern.
∴ Volume of water absorbed by ‘n’ NCERT Solutions for Class 10 Maths Chapter 12 - Surface Areas and Volumes (Exercise 12.5)
∴ [Volume occupied by ‘n’ bricks] = [(free space in the cistern) + (volume of water absorbed by n-bricks)]
NCERT Solutions for Class 10 Maths Chapter 12 - Surface Areas and Volumes (Exercise 12.5)
NCERT Solutions for Class 10 Maths Chapter 12 - Surface Areas and Volumes (Exercise 12.5)
Thus, 1792 bricks can be put in the cistern.

Q 4. In one fortnight of a given month, there was a rainfall of 10 cm in a river valley. If the area of the valley is 7280 km3, show that the total rainfall was approximately equivalent to the addition to the normal water of three rivers each 1072 km long, 75 m wide and 3 m deep.
Sol. Volume of three rivers = 3 {(Surface area of a river) × Depth}
NCERT Solutions for Class 10 Maths Chapter 12 - Surface Areas and Volumes (Exercise 12.5)
Volume of rainfall = (surface area) × (height of rainfall)
NCERT Solutions for Class 10 Maths Chapter 12 - Surface Areas and Volumes (Exercise 12.5)
NCERT Solutions for Class 10 Maths Chapter 12 - Surface Areas and Volumes (Exercise 12.5)
Since, 0.7236 km3 ≠ 9.728 km3
∴ The additional water in the three rivers is not equivalent to the rainfall.

Q 5. An oil funnel made of tin sheet consists of a 10 cm long cylindrical portion attached to a frustum of a cone. If the total height is 22 cm, diameter of the cylindrical portion is 8 cm and the diameter of the top of the funnel is 18 cm, find the area of the tin sheet required to make the funnel (see Fig.).
NCERT Solutions for Class 10 Maths Chapter 12 - Surface Areas and Volumes (Exercise 12.5)
Sol. We have, for the cylindrical part
Diameter = 8 cm
⇒ Radius (r) = 4 cm
Height = 10 cm
⇒ Curved Surface area  NCERT Solutions for Class 10 Maths Chapter 12 - Surface Areas and Volumes (Exercise 12.5)
For the frustum:  NCERT Solutions for Class 10 Maths Chapter 12 - Surface Areas and Volumes (Exercise 12.5)
and  NCERT Solutions for Class 10 Maths Chapter 12 - Surface Areas and Volumes (Exercise 12.5)
Height (H) = 22 − 10 = 12 cm
∴ Slant height (l) NCERT Solutions for Class 10 Maths Chapter 12 - Surface Areas and Volumes (Exercise 12.5)
NCERT Solutions for Class 10 Maths Chapter 12 - Surface Areas and Volumes (Exercise 12.5)
∴Surface area,  NCERT Solutions for Class 10 Maths Chapter 12 - Surface Areas and Volumes (Exercise 12.5)
NCERT Solutions for Class 10 Maths Chapter 12 - Surface Areas and Volumes (Exercise 12.5)
Area of tin required = [Area of the frustum] + [Area of cylindrical portion]
NCERT Solutions for Class 10 Maths Chapter 12 - Surface Areas and Volumes (Exercise 12.5)

Q 6. Derive the formula for the curved surface area and total surface area of the frustum of a cone, given to you in Section 13.5, using the symbols as explained.
NCERT Solutions for Class 10 Maths Chapter 12 - Surface Areas and Volumes (Exercise 12.5)
Sol. We have,
Curved surface area of the frustum PQRS
NCERT Solutions for Class 10 Maths Chapter 12 - Surface Areas and Volumes (Exercise 12.5)
= πr1 l1 − πr2 l2      ...(1)
Now,     ΔOC1 Q ~ ΔOC2S
NCERT Solutions for Class 10 Maths Chapter 12 - Surface Areas and Volumes (Exercise 12.5)
NCERT Solutions for Class 10 Maths Chapter 12 - Surface Areas and Volumes (Exercise 12.5)
NCERT Solutions for Class 10 Maths Chapter 12 - Surface Areas and Volumes (Exercise 12.5)NCERT Solutions for Class 10 Maths Chapter 12 - Surface Areas and Volumes (Exercise 12.5)
NCERT Solutions for Class 10 Maths Chapter 12 - Surface Areas and Volumes (Exercise 12.5)
NCERT Solutions for Class 10 Maths Chapter 12 - Surface Areas and Volumes (Exercise 12.5)
Now, from (1), curved surface area of the frustum
NCERT Solutions for Class 10 Maths Chapter 12 - Surface Areas and Volumes (Exercise 12.5)
NCERT Solutions for Class 10 Maths Chapter 12 - Surface Areas and Volumes (Exercise 12.5)     NCERT Solutions for Class 10 Maths Chapter 12 - Surface Areas and Volumes (Exercise 12.5)
Now, the total surface area of the frustum = (curved surface area) + (base surface area) + (top surfaces area)
NCERT Solutions for Class 10 Maths Chapter 12 - Surface Areas and Volumes (Exercise 12.5)

Q 7. Derive the formula for the volume of the frustum of a cone, given to you in Section 13.5, using the symbols as explained.
Sol. We have, [Volume of the frustum RPQS]  = [Volume of right circular cone OPQ] − [Volume of right circular cone ORS]
NCERT Solutions for Class 10 Maths Chapter 12 - Surface Areas and Volumes (Exercise 12.5)

NCERT Solutions for Class 10 Maths Chapter 12 - Surface Areas and Volumes (Exercise 12.5)
Since ΔOC1Q ~ ΔOC2S
NCERT Solutions for Class 10 Maths Chapter 12 - Surface Areas and Volumes (Exercise 12.5)
NCERT Solutions for Class 10 Maths Chapter 12 - Surface Areas and Volumes (Exercise 12.5)
NCERT Solutions for Class 10 Maths Chapter 12 - Surface Areas and Volumes (Exercise 12.5)
From (1) and (2), we have
{volume of the frustum RPQS}  NCERT Solutions for Class 10 Maths Chapter 12 - Surface Areas and Volumes (Exercise 12.5)
NCERT Solutions for Class 10 Maths Chapter 12 - Surface Areas and Volumes (Exercise 12.5)

The document NCERT Solutions for Class 10 Maths Chapter 12 - Surface Areas and Volumes (Exercise 12.5) is a part of the Bank Exams Course NCERT Mathematics for Competitive Exams.
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FAQs on NCERT Solutions for Class 10 Maths Chapter 12 - Surface Areas and Volumes (Exercise 12.5)

1. What is the formula for finding the surface area of a cube?
Ans. The formula for finding the surface area of a cube is 6a^2, where 'a' represents the length of the side of the cube.
2. How do you calculate the volume of a cylinder?
Ans. The volume of a cylinder can be calculated using the formula V = πr^2h, where 'r' represents the radius of the base and 'h' represents the height of the cylinder.
3. How can I find the surface area of a cone?
Ans. The surface area of a cone can be found using the formula A = πr(r + l), where 'r' represents the radius of the base and 'l' represents the slant height of the cone.
4. How do you determine the volume of a sphere?
Ans. The volume of a sphere can be determined using the formula V = (4/3)πr^3, where 'r' represents the radius of the sphere.
5. What is the formula for finding the lateral surface area of a cylinder?
Ans. The formula for finding the lateral surface area of a cylinder is 2πrh, where 'r' represents the radius of the base and 'h' represents the height of the cylinder.
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