Advertisements
Advertisements
Question
A solid sphere and a solid hemi-sphere have the same total surface area. Find the ratio between their volumes.
Solution
Let the radius of the sphere be 'r1'.
Let the radius of the hemisphere be 'r2'
TSA of sphere = `4pi r_1^2`
TSA of hemisphere = `3pi r_2^2`
TSA of sphere = TSA of hemi-sphere
`4pir_1^2 = 3pir_2^2`
`=> r_2^2 = 4/3r_1^2`
`=> r_2 = 2/sqrt3r_1`
Volume of sphere, `v_1 = 4/3pir_1^3`
Volume of hemisphere, `v_2 = 2/3pir_2^3`
`v_2 = 2/3pir_2^3`
`=> v_2 = 2/3pi((r_1 2)/(3sqrt3))^3`
`=> v_2 = 2/3pi(r_2^3 8)/(3sqrt3)`
Dividing v1 by v2
`v_1/v_2 = (4/3pir_1^3)/(2/3pi 8/(3sqrt3)r_1^3`
`=> v_1/v_2 = (4/3)/(2/3 8/(3sqrt3)`
`=> v_1/v_2 = 4/3 xx (9sqrt3)/16`
`=> v_1/v_2 = (3sqrt3)/4`
APPEARS IN
RELATED QUESTIONS
Find the curved surface area of a cone, if its slant height is 60 cm and the radius of its base is 21 cm.
The height of a cone is 21 cm. Find the area of the base if the slant height is 28 cm.
Find the ratio of the curved surface areas of two cones if their diameters of the bases are equal and slant heights are in the ratio 4 : 3.
A conical tent is 10 m high and the radius of its base is 24 m. Find the slant height of the tent. If the cost of 1 2 m canvas is Rs. 70, find the cost of the canvas required to make the tent.
A heap of wheat is in the form of a cone of diameter 9 m and height 3.5 m. Find its volume. How much canvas cloth is required to just cover the heap? (Use ЁЭЬЛ = 3.14).
Monica has a piece of Canvas whose area is 551 m2. She uses it to have a conical tent made, with a base radius of 7m. Assuming that all the stitching margins and wastage incurred while cutting, amounts to approximately 1 m2. Find the volume of the tent that can be made with it.
Find the volume of a cone whose slant height is 17 cm and radius of base is 8 cm.
The area of the base of a conical solid is 38.5 cm2 and its volume is 154 cm3. Find the curved surface area of the solid.
A vessel, in the form of an inverted cone, is filled with water to the brim. Its height is 32 cm and diameter of the base is 25.2 cm. Six equal solid cones are dropped in it, so that they are fully submerged. As a result, one-fourth of water in the original cone overflows. What is the volume of each of the solid cones submerged?
The internal and external diameter of a hollow hemispherical vessel are 21 cm and 28 cm respectively. Find :
- internal curved surface area,
- external curved surface area,
- total surface area,
- volume of material of the vessel.
A cubical block of side 7 cm is surmounted by a hemisphere of the largest size. Find the surface area of the resulting solid.
The radii of the bases of two solid right circular cones of same height are r1 and r2 respectively. The cones are melted and recast into a solid sphere of radius R. Find the height of each cone in terms r1, r2 and R.
The cross-section of a railway tunnel is a rectangle 6 m broad and 8 m high surmounted by a semi-circle as shown in the figure. The tunnel is 35 m long. Find the cost of plastering the internal surface of the tunnel (excluding the floor) at the rate of Rs. 2.25 per m2.
The given figure shows the cross section of a water channel consisting of a rectangle and a semi-circle. Assuming that the channel is always full, find the volume of water discharged through it in one minute if water is flowing at the rate of 20 cm per second. Give your answer in cubic metres correct to one place of decimal.
Find the radius of the circular base of the cone , if its volume is 154 cm3 and the perpendicular height is 12 cm
The radius and height of a cylinder, a cone and a sphere are same. Calculate the ratio of their volumes.
The given figure shows the cross-section of a cone, a cylinder and a hemisphere all with the same diameter 10 cm and the other dimensions are as shown. Calculate: the density of the material if its total weight is 1.7 kg
The ratio of the base area and the curved surface of a conical tent is 40: 41. If the height is 18 m, Find the air capacity of the tent in terms of n.
A conical tent is accommodate to 11 persons each person must have 4 sq. metre of the space on the ground and 20 cubic metre of air to breath. Find the height of the cone.
How many square metres of canvas is required for a conical tent whose height is 3.5 m and the radius of the base is 12 m?
