Mensuration 3D - Study Mode

[#121] The radius and height of a right circular cone are in the ratio 3 : 4. If its curved surface area (in cm 2 ) is 240π. Then its volume (in cm 3 ) is:
Correct Answer

(D) 768π

Explanation

Solution: r : h = 3 : 4 $$l$$ : r = 5 : 3 $$eqalign{
& { ext{Curved surface area}} = pi rl cr
& 240pi = pi imes 3x imes 5x cr
& {x^2} = 16 cr
& x = 4 cr
& herefore r = 3 imes 4 = 12 cr
& { ext{Volume}} = frac{1}{3}pi {r^2}h cr
& = frac{1}{3}pi imes 144 imes 16 cr
& = 768pi cr} $$

[#122] A cylindrical pencil of diameter 1.2 cm has one of its end sharpened into a conical shape of height 1.4 cm. The volume of the material removed is
Correct Answer

(C) 1.056 cm 3

Explanation

Solution: $$eqalign{
& { ext{Required volume removed}} cr
& = pi {r^2}h - frac{1}{3}pi {r^2}h cr
& = frac{2}{3}pi {r^2}h cr
& = frac{2}{3} imes frac{{22}}{7} imes .6 imes .6 imes 1.4 cr
& = 1.056{ ext{ c}}{{ ext{m}}^3} cr} $$

[#123] A sphere and a hemisphere have the same volume. The ratio of their curved surface area is:
Correct Answer

(D) $${2^{frac{1}{3}}}:1$$

Explanation

Solution: Let the radius of hemisphere and sphere be 'r' and 'R' $$eqalign{
& Rightarrow frac{4}{3}pi {R^3} = frac{2}{3}pi {r^3} cr
& frac{{{R^3}}}{{{r^3}}} = frac{1}{2} cr
& frac{R}{r} = frac{1}{{
oot 3 of 2 }} cr
& Rightarrow { ext{Ratio of curved surface area}} cr
& = frac{{4pi {R^2}}}{{2pi {r^2}}} cr
& = frac{{2{R^2}}}{{{r^2}}} cr
& = frac{{2 imes 1}}{{{{left( {
oot 3 of 2 }
ight)}^2}}} cr
& = frac{2}{{{{left( 2
ight)}^{frac{2}{3}}}}} cr
& Rightarrow frac{R}{r} = frac{{{2^{frac{1}{3}}}}}{1} cr} $$

[#124] A tank 40 m long, 30 m broad and 12 m deep is dug in a field 1000 m long and 30 m wide. By how much will the level of the field rise if the earth dug out of the tank is evenly spread over the field?
Correct Answer

(C) 0.5 metre

Explanation

Solution: Volume of earth taken out = 40 × 30 × 12 = 14400 m 3 Area of rectangular field = 1000 × 30 = 30000 m 2 Area of region of tank = 40 × 30 = 1200 m 2 Remaining area = 30000 - 1200 = 28800 m 2 Increase in height $$ = frac{{14400}}{{28800}} = 0.5{ ext{ m}}$$

[#125] A tank is in the form of a cuboid with length 12 m. If 18 kilolitre of water is removed from it, the water level goes down by 30 cm. What is the width (in m) of the tank?
Correct Answer

(A) 5

Explanation

Solution: Volume of cuboid = $$l$$bh 12 × b × h - 18 = 12 × b × (h - 0.3) 12bh - 18 = 12bh - 3.6b b = 5 m