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प्रश्न
A right circular cylinder and a right circular cone have equal bases and equal heights. If their curved surfaces are in the ratio 8 : 5, determine the ratio of the radius of the base to the height of either of them.
उत्तर
For right circular cylinder, let r1 = r, h1 = h.
Then, curved surface area, s1 of cylinder = `2pir_1 = 2pirh ........... (1)`
For right circular cone, let r2 = r, h2 = h
Then, curved surface area, s2 of cone = `pi r_2l " where "l = sqrt(r_2^2 + h_2^2) = sqrt(r^2 + h^2)`
` = pir sqrt(r^2 + h^2) ..................(2)`
Divide (i) and (ii),
`s_1/s_2 = (2pirh)/(pirsqrt(r^2 + h^2))`
`8/5 = (2h)/sqrt(r^2 + h^2) [s_1/s_2 = 8/5]`
`64/25 = (4h^2)/(r^2 + h^2)` [squaring]
`64r^2 + 64h^2 = 100h^2`
`64r^2 = 36h^2`
`16r^2 = 9h^2`
`r^2 / h^2 = 9/16`
`r/h = 3/4`
`therefore r : h = 3:4`
संबंधित प्रश्न
A sphere of diameter 12 cm, is dropped in a right circular cylindrical vessel, partly filled with water. If the sphere is completely submerged in water, the water level in the cylindrical vessel rises by 3 `5/9` cm. Find the diameter of the cylindrical vessel.
Due to heavy floods in a state, thousands were rendered homeless. 50 schools collectively offered to the state government to provide place and the canvas for 1500 tents to be fixed by the governments and decided to share the whole expenditure equally. The lower part of each tent is cylindrical of base radius 2.8 cm and height 3.5 m, with conical upper part of same base radius but of height 2.1 m. If the canvas used to make the tents costs Rs. 120 per sq. m, find the amount shared by each school to set up the tents. What value is generated by the above problem? (use `pi =22/7`)
Volume and surface area of a solid hemisphere are numerically equal. What is the diameter of hemisphere?
A solid sphere of diameter 6 cm is dropped in a right circular cylindrical vessel partly filled with water. The diameter of the cylindrical vessel is 12 cm. If the sphere is completely submerged in water, by how much will the level of water rise in the cylindrical vessel?
The radii of two cylinders are in the ratio 3 : 5 and their heights are in the ratio 2 : 3. What is the ratio of their curved surface areas?
A hemispherical bowl of internal radius 9 cm is full of water. Its contents are emptied into a cylindrical vessel of internal radius 6 cm. Find the height of water in the cylindrical vessel.
The sum of length, breadth and height of a cuboid is 19 cm and its diagonal is `5sqrt(5)` cm. Its surface area is
A medicine capsule is in the shape of a cylinder of diameter 0.5 cm with two hemispheres stuck to each of its ends. The length of entire capsule is 2 cm. The capacity of the capsule is ______.
Match the column:
| (1) surface area of cuboid | (A) πr2h |
| (2) surface area of closed right cylinder | (B) 2πr(h + r) |
| (3) Total surface area of right cone | (C) πrl + πr2 |
| (4) Total surface area of hemisphere | (D) 3πr3 |
| (E) 3πr2 | |
| (F) 2[lb + bh + lh] |
The radius of a metal sphere is 3 cm. The sphere is melted and made into a long wire of uniform circular cross-section, whose length is 36 cm. To calculate the radius of wire, complete the following activity.
Radius of the sphere = `square`
Length of the wire = `square`
Let the radius of the wire by r cm.
Now, Volume of the wire = Volume of the `square`
`square` = `square`
r2 × `square` = `square` × `square`
r2 × `square` = `square`
r = `square`
Hence, the radius of the wire is `square` cm.
