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1.

If the length of the radii of two solid right circular cylinder are in the ratio 2:3 and their heights are in the ratio 5:3 , then the ratio of their lateral surfaces isA. `2:5`B. `8:7`C. `10:9`D. `16:9`

Answer» Correct Answer - C
Let the radii of two solid right circular cylinders are 2x units and 3x units, such that their ratio is `2x:3x=2:3`, which satisfies the given condition. Also let their heights be 5 h units and 3h units respectively.
`:.` the area of the lateral surface of the first `=2pi xx 2x xx 5h` sq-units `=20 pi x h ` sq-units.
and the area of the lateral surface of the second `=2pi xx 3x xx 3h` sq-units `=18pi x h ` sq-units.
Hence the required ratio `=2pi xx h : 18 pi x h = 10:9`
`:.` (c) is correct.
Discussion
2.

If the length of the radius of a right circular cylinder is doubled and height is halved, then its lateral surface area will beA. equalB. doubleC. halfD. 3 times

Answer» Correct Answer - A
Let the radius of the cylinder be r units and height of it be h units.
`:.` the lateral surface area `= 2 pi rh ` sq-units
If the radius is doubled, i.e., 2r units and the height is halved, i.e., `h/2` units, then the lateral surface area becomes `2 pi xx 2r xx h/2 sq - "units" = 2pi r h ` sq-units.
`:.` The lateral surface area remains the same , i.e., in both the cases the lateral surface areas are equal.
`:.` (a) is correct.
Discussion
3.

If volumes of two solid right circular cylinders are same and their heights are in the ratio 1:2, then the ratio of lengths of their radii is -A. `1 : sqrt(2)`B. `sqrt(2):1`C. `1:2`D. `2:1`

Answer» Correct Answer - B
Let the heights of the cylinders are h units and 2h units ( since the ratio is 1:2 ) and let the radii of the cylinders are R and r units respectively.
Since the volume of the cylinders are same,
`:. pi R^(2) h = pi r^(2) .2h rArr R^(2)=2r^(2) rArr sqrt(R^(2))=sqrt(2r^(2))` [ Taking square root of both the sides ]
`rArr R=sqrt(2) r rArr (R)/(r) = sqrt(2) rArr R:r = sqrt(2) :1`
Hence the required ratio `=sqrt(2):1`
`:.` (b) is correct.
Discussion
4.

Rahamal got a playing top (lattu) as his birthday presentation, which surprisingly had no colour on it. So he wanted to colour it with his crayons. The top is shaped like a cone surprisingly by a hemisphere. The entire top is 5 cm in height and the diameter of the top is 3.5 cm. Find the area he has to colour.

Answer» Correct Answer - 39.6 sq-cm ( approx)
Total surface area of the top = curved surface area of the hemisphere + curved surface area of the cone.
Now, the curved surface area of the hemisphere `=1/2 xx 4 pi r^(2) = 2 pi r^(2) = (2xx(22)/(7)xx(3.5)/(2)xx(3.5)/(2))`sq-cm
Also, the height of the cone = height of the top - height ( radius) of the hemispherical part.
`=(5-(3.5)/(2)) cm = 3.25`cm.
So the slant height of the cone `=sqrt((3.5/2)^(2)+(3.25)^(2))cm=3.7cm` (approx)
`:.` curved surface area of cone `=((22)/(7)xx(3.5)/(2)xx3.7)`sq-cm [ Formula `: pi r l` ]
Hence the total surface area of the top
`=(2xx(22)/(7)xx(3.5)/(2)xx(3.5)/(2)) sq-cm + ((22)/(7)xx(3.5)/(2)xx3.7)sq-cm.=39.6 sq-cm `[(approx.)]
Discussion