Volume And Surface Area - Study Mode
[#51] A right circular cylinder and a sphere are of equal volumes and their radii are also equal. If h is the height of the cylinder and d, the diameter of the sphere, then :
Correct Answer
(D) $$frac{h}{2} = frac{d}{3}$$
Explanation
Solution: Let the radius of the sphere and that of the right circular cylinder be r Then, Volume of the cylinder $$ = pi {r^2}h$$ Volume of the sphere $$ = frac{4}{3}pi {r^3}$$ $$eqalign{
& herefore pi {r^2}h = frac{4}{3}pi {r^3} cr
& Rightarrow 3h = 4r cr
& Rightarrow 3h = 2d cr
& Rightarrow frac{h}{2} = frac{d}{3} cr} $$
[#52] A sphere of maximum volume is cut out from a solid hemisphere of radius r. The ratio of the volume of the hemisphere to that of the cut out sphere is :
Correct Answer
(B) 4 : 1
Explanation
Solution: Volume of hemisphere = $$frac{2}{3}pi {r^3}$$ Volume of biggest sphere : = Volume of sphere with diameter r $$eqalign{
& = frac{4}{3}pi {left( {frac{r}{2}}
ight)^3} cr
& = frac{1}{6}pi {r^3} cr} $$ ∴ Required ratio : $$eqalign{
& = frac{{frac{2}{3}pi {r^3}}}{{frac{1}{6}pi {r^3}}} cr
& = frac{4}{1}i.e.,4:1 cr} $$
[#53] A plot of land in the form of a rectangle has dimensions 240 m × 180 m. A drain-let 10 m wide is dug all around it (outside) and the earth dug out is evenly spread over the plot, increasing its surface level by 25 cm. The depth of the drain-let is :
Correct Answer
(C) 1.227 m
Explanation
Solution: Volume of earth dug out : = (240 × 180 × 0.25) m 3 = 10800 m 3 Let the depth of the drain-let be h metres Then, Volume of earth dug out : = [{(260 × 200) - (240 × 180)}h ] m 3 = (8800h) m 3 ∴ 8800h = 10800 ⇒ h = $$frac{10800}{8800}$$ ⇒ h = $$frac{27}{22}$$ ⇒ h = 1.227 m
[#54] A 4 cm cube is cut into 1 cm cubes. The total surface area of all the small cubes is :
Correct Answer
(D) None of these
Explanation
Solution: Number of small cube formed : $$eqalign{
& = left( {frac{{4 imes 4 imes 4}}{{1 imes 1 imes 1}}}
ight) cr
& = 64 cr} $$ Total surface area of the small cubes : $$eqalign{
& = left[ {64 imes left( {6 imes {1^2}}
ight)}
ight]{ ext{c}}{{ ext{m}}^2} cr
& = 384{ ext{ c}}{{ ext{m}}^2} cr} $$
[#55] A well has to be dug out that is to be 22.5 m deep and of diameter 7 m. Find the cost of plastering the inner curved surface at Rs. 3 per sq.meter :
Correct Answer
(C) Rs. 1485
Explanation
Solution: Curved surface area : $$eqalign{
& = 2pi rh cr
& = left( {2 imes frac{{22}}{7} imes frac{7}{2} imes 22.5}
ight){{ ext{m}}^2} cr
& = 495{ ext{ }}{{ ext{m}}^2} cr} $$ ∴ Cost of plastering : = Rs. (495 × 3) = Rs. 1485