Advertisements
Advertisements
Question
A hemispherical depression is cut out from one face of a cubical wooden block such that the diameter l of the hemisphere is equal to the edge of the cube. Determine the surface area of the remaining solid.
Advertisements
Solution
Diameter of hemisphere = Edge of cube = l
Radius of hemisphere = `l/2`
Curved surface area of hemisphere = 2πr2
= `2 xx pi xx l/2 xxl/2 xx (pi l^2)/2`
Base area of the hemisphare = πr2
= `pi (l/2)^2 = (pil^2)/4`
Surface area of the cube = `6 xx l^2 = 6l^2`
Surface area of the remaining solid = [Total surface area of cube + C.S.A. of hemispjere − base area of hemisphere]
= `6l^2 + (pil^2)/2 − (pil^2)/2`
= `(24l^2 + 2pil^2 − pil^2)/4`
= `(24l^2 + pil^2)/4`
= `l^2/4 (24 + pi)` sq. units.
APPEARS IN
RELATED QUESTIONS
150 spherical marbles, each of diameter 1.4 cm, are dropped in a cylindrical vessel of diameter 7 cm containing some water, which are completely immersed in water. Find the rise in the level of water in the vessel.
A vessel is in the form of a hollow hemisphere mounted by a hollow cylinder. The diameter of the hemisphere is 14 cm and the total height of the vessel is 13 cm. Find the inner surface area of the vessel. [Use `pi = 22/7`]
A toy is in the form of a cone of radius 3.5 cm mounted on a hemisphere of same radius. The total height of the toy is 15.5 cm. Find the total surface area of the toy [Use π =`22/7`]
The diameters of the lower and upper ends of a bucket in the form of a frustum of a cone are 10 cm and 30 cm respectively. If its height is 24 cm, find:
1) The area of the metal sheet used to make the bucket.
2) Why we should avoid the bucket made by ordinary plastic? [Use π = 3.14]
A tent consists of a frustum of a cone capped by a cone. If the radii of the ends of the frustum be 13 m and 7 m , the height of the frustum be 8 m and the slant height of the conical cap be 12 m, find the canvas required for the tent. (Take : π = 22/7)
Radii of circular ends of a solid frustum off a cone re 33cm and 27cm and its slant height are 10cm. find its total surface area?
The largest cone is curved out from one face of solid cube of side 21 cm. Find the volume of the remaining solid.
If r1 and r2 be the radii of two solid metallic spheres and if they are melted into one solid sphere, prove that the radius of the new sphere is \[\left( r_1^3 + r_2^3 \right)^\frac{1}{3}\].
Find the volume of a solid in the form of a right circular cylinder with hemi-spherical ends whose total length is 2.7 m and the diameter of each hemi-spherical end is 0.7 m.
If the total surface area of a solid hemisphere is 462 cm2, find its volume.
A spherical ball of radius 3 cm is melted and recast into three spherical balls. The radii of two of these balls are 1.5 cm and 2 cm. Find the radius of the third ball.
Water is flowing through a cylindrical pipe of internal diameter 2 cm, into a cylindrical tank of base radius 40 cm, at the rate of 0.4 m per second. Determine the rise in level of water in the tank in half an hour.
A copper wire of diameter 6 mm is evenly wrapped on a cylinder of length 18 cm and diameter 49 cm to cover its whole surface. Find the length and the volume of the wire. If the density of copper be 8.8 g per cm3, then find the weight of the wire.
A plumbline (sahul) is the combination of ______.
The volume of a hemisphere is 19404 cm3. The total surface area of the hemisphere is ______.
If the surface areas of two spheres are in ratio 16 : 9, then their volumes will be in the ratio ______.
Eight solid sphere of same size are made by melting a solid metallic cylinder of base diameter 6 cm and height 32 cm. The diameter of each sphere is ______.
A solid cone of radius r and height h is placed over a solid cylinder having same base radius and height as that of a cone. The total surface area of the combined solid is `pir [sqrt(r^2 + h^2) + 3r + 2h]`.
3 cubes each of 8 cm edge are joined end to end. Find the total surface area of the cuboid.
