Mensuration 3D - Study Mode

[#116] The radius of the base of a cylinder is 14 cm and its volume is 6160 cm 3 . The curved surface area (in cm 2 ) is: $$left( {pi = frac{{22}}{7}}
ight)$$
Correct Answer

(A) 880

Explanation

Solution: r = 14 cm πr 2 h = 6160 cm 3 Curved surface area = ? $$frac{{22}}{7}$$ × 14 × 14 × h = 6160 h = 10 Curved surface area = 2πrh = 2 × $$frac{{22}}{7}$$ × 14 × 10 = 880 cm 2

[#117] A solid metallic sphere of radius 15 cm is melted and recast into spherical balls of radius 3 cm each. What is the ratio of the surface area of the original sphere and the sum of the surface areas of all balls?
Correct Answer

(A) 1 : 5

Explanation

Solution: R 3 = nr 3 15 × 15 × 15 = n × 3 × 3 × 3 n = 125 (n = number of small spherical balls) $$eqalign{
& frac{{{S_1}}}{{{S_2}}} = frac{{{R^2}}}{{n{r^2}}} cr
& frac{{{S_1}}}{{{S_2}}} = frac{{15 imes 15}}{{125 imes 3 imes 3}} cr
& {S_1}:{S_2} = 1:5 cr} $$

[#118] Two solid right cones of equal height and of radii r 1 and r 2 are melted and made to form a solid sphere of radius R. Then the height of the cone is
Correct Answer

(C) $$frac{{4{R^3}}}{{r_1^2 + r_2^2}}$$

Explanation

Solution: $$eqalign{
& { ext{Let the height be }}H cr
& Rightarrow frac{1}{3}pi r_1^2H + frac{1}{3}pi r_2^2H = frac{4}{3}pi {R^3} cr
& Rightarrow frac{1}{3}pi Hleft( {r_1^2 + r_2^2}
ight) = frac{4}{3}pi {R^3} cr
& Rightarrow H = frac{{4{R^3}}}{{r_1^2 + r_2^2}} cr} $$

[#119] The curved surface area of a right cylinder is 3696 cm 2 . Its height is three times its radius. What is the capacity (in liters) of the cylinder? $$left( {{ ext{Take }}pi = frac{{22}}{7}}
ight)$$
Correct Answer

(A) 25.872

Explanation

Solution: $$eqalign{
& r:h = 1:3 cr
& 2pi rh = 3696 cr
& 2 imes frac{{22}}{7} imes x imes 3x = 3696 cr
& 3{x^2} = frac{{3696 imes 7}}{{44}} cr
& {x^2} = frac{{1232 imes 7}}{{44}} cr
& {x^2} = frac{{112 imes 7}}{4} cr
& {x^2} = 28 imes 7 cr
& x = 14 cr
& { ext{Volume}} = pi {r^2}h cr
& = frac{{22}}{7} imes {14^2} imes 42 cr
& = frac{{22}}{7} imes 14 imes 14 imes 42 cr
& = 25872{ ext{ c}}{{ ext{m}}^3} cr
& = 25.872{ ext{ L}} cr} $$

[#120] Water flows into a tank which is 200 m long and 150 m wide through a pipe of cross-section 0.3 m × 0.2 m at 20 km/hour. Then the time (in hours) for the water level in the tank to reach 8 m is
Correct Answer

(D) 200

Explanation

Solution: Let the number of hours be x ⇒ (0.3 × 0.2 × 20000) × x = 200 × 150 × 8 ⇒ x = $$frac{{200 imes 150 imes 8}}{{3 imes 2 imes 200}}$$ ⇒ x = 200 hrs.