Chapter 11 MENSURATION
Exercise 11.4
Q.1 Given a cylindrical tank, in which situation will you find surface area and in which situation volume.
(a) To find how much it can hold.
(b) Number of cement bags required to plaster it.
(c) To find the number of smaller tanks that can be filled with water from it.
Sol. (a) To find how much it can hold we will need to find the volume.
(b) To find the number of cement bags required to plaster it we will need to find the surface area.
(c) To find the number of smaller tanks that can be filled with water from it we will need to find the volume.
Q.2 Diameter of cylinder A is 7 cm, and the height is 14 cm. Diameter of cylinder B is 14 cm and height is 7 cm. Without doing any calculations can you suggest whose volume is greater? Verify it by finding the volume of both the cylinders. Check whether the cylinder with greater volume also has greater surface area?
Sol. Without calculations, we can say that volume of cylinder B is greater than the cylinder A, as the radius of the cylinder B is greater than the cylinder A and volume of is cylinder is proportional to the square of the radius.
Calculation for cylinder A:
Given, diameter (d) = 7 cm and height (h) = 14 cm
Radius of cylinder (r) = d/2 = 7/2 cm
Therefore, volume of cylinder A = Ï€r^{2}h
= 22/7 x 7/2 x 7/2 x 14
= 539 cm^{2}
Surface area of cylinder A =Â 2Ï€r(r+h)
= [2 x 22/7 / 7/2(7/2+14)] cm^{2}
=Â [2 x (7+28/2)] cm^{2}
=Â [2 x (35/2)] cm^{2}
=Â Â 385 cm^{2}
Calculation for cylinder B:
Given, diameter (d) = 14 cm and height (h) = 7 cm
Radius of cylinder (r) = d/2 = 14/2 = 7cm
Therefore, volume of cylinder B =Â Ï€r^{2}h
= 22/7 x 7 x 7 x 7
= 1078 cm^{2}
Surface area of cylinder B = 2Ï€r(r+h)
= [2 x 22/7 x 7(7+7)]cm^{2}
=Â [44 x 14]cm^{2}
=616 cm^{2}
After calculations, we can see that the volume of cylinder B is greater than that of cylinder A. Also, the cylinder B with greater volume has the greater surface area.
Q.3 Find the height of a cuboid whose base area is 180 cm^{2} and volume is 900 cm^{3}?
Sol. Given, base area of cuboid = 180 cm^{2} and volume of cuboid is 900 cm^{3}
Now, volume of cuboid = l x b x h
900 = 180 x h
h = 900/180 = 5 m
Hence, the height of cuboid is 5 m.
Q.4 A cuboid is of dimensions 60 cm Ã— 54 cm Ã— 30 cm. How many small cubes with side 6 cm can be placed in the given cuboid?
Sol. Given, volume of cuboid = 60 cm Ã— 54 cm Ã— 30 cm = 97200 cm^{3 }and side of cube = 6 cm
Now, volume of cube = (6)^{3} cm^{3} = 216 cm^{3}
Therefore, required number of cubes = Volume of the cuboid / Volume of the cube
= 97200/216
= 450
Hence, 450 number of cubes can be placed in the given cuboid.
Q.5 Find the height of the cylinder whose volume is 1.54 m^{3} and diameter of the base is 140 cm?
Sol. Given, volume of cylinder = 1.54 m^{3} and diameter of base (d) = 140 cm
Here, radius of the cylinder (r) = d/2 = 140/2 = 70 cm = 70/100 m
Now, volume of cylinder =Â Ï€r^{2}h
1.54 m^{3} = 22/7 x 70/100 m x 70/100 m x h
h = (1.54x100 / 22x7)m = 1 m
Hence, the height of the cylinder is 1 m.
Q.6 A milk tank is in the form of cylinder whose radius is 1.5 m and length is 7 m. Find the quantity of milk in litres that can be stored in the tank?
Sol. Given, radius of cylinder (r) = 1.5 m and length of cylinder (h) = 7 m
Now, volume of cylinder =Â Ï€r^{2}h
= 22/7 x 1.5 x 1.5 x 7 m^{3}
=Â 49.5 m^{3}
= 49.5 x 1000 litres (Since, 1 m^{3} = 1000 litres)
= 49500 litres
Hence, the quantity of milk that can be stored in the tank is 49500 litres.
Q.7 If each edge of a cube is doubled,
(i) how many times will its surface area increase?
(ii) how many times will its volume increase?
Sol. (i) Let the original edge of the cube be l.
Therefore, original surface area = 6l^{2}
Now, edge of cube is doubled, hence it will become 2l.
Therefore, new surface area = 6(2l)^{2} = 24l^{2} = 4 x 6l^{2}
Hence, if each edge of cube is doubled then its surface area will increase by 4 times.
(ii) Let the original edge of the cube be l.
Therefore, original volume = l^{3}
Now, edge of cube is doubled, hence it will become 2l.
Therefore, new surface area = (2l)^{3} = 8l^{3} = 8 x l^{3}
Hence, if each edge of cube is doubled then its volume will increase by 8 times.
Q.8 Water is pouring into a cubiodal reservoir at the rate of 60 litres per minute. If the volume of reservoir is 108 m^{3}, find the number of hours it will take to fill the reservoir.
Sol. Given, volume of reservoir = 108 m^{3}
Also, given water is pouring into cuboidal reservoir at the rate of 60 litres per minute which means (60 x 60)L = 3600 L per hour
Therefore, required number of hours = 108000/3600 = 30 hours
Hence, it will take 30 hours to fill the reservoir.
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