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Question
From a solid cylinder whose height is 8 cm and radius 6 cm, a conical cavity of height 8 cm and base radius 6 cm is hollowed out. Find the volume of the remaining solid. Also, find the total surface area of the remaining solid.
Solution 1
Volume of the solid left = Volume of cylinder - Volume of cone `=pir^2h - 1/3pir^2h = 2/3 xx 22/7 xx 8 xx 6 xx 6 = 603.428 "cm^3'`
The slant length of the cone , `l =sqrt(r^2 + h^2) = sqrt(36+64) = 10 "cm"`
Total surface area of final solid = Area of base circle
Solution 2
Volume of the solid left = Volume of cylinder - Volume of cone `=pir^2h - 1/3pir^2h = 2/3 xx 22/7 xx 8 xx 6 xx 6 = 603.428 "cm^3'`
The slant length of the cone ,`l =sqrt(r^2 + h^2) = sqrt(36+64) = 10 "cm"`
Total surface area of final solid = Area of base circle + Curved surface area of cylinder +curved area of once`= pirl^2 + 2pirh + pirl = pir (r + 2h +l) = 22/7 xx 6 xx (6 + 16 + 10) = 603.42 "cm"^2`
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Assertion (A)
If the volumes of two spheres are in the ratio 27 : 8, then their surface areas are in the ratio 3 : 2.
Reason (R)
Volume of a sphere `=4/3pi"R"^3`
Surface area of a sphere = 4πR2
Both Assertion (A) and Reason (R) are true and Reason (R) is a correct explanation of Assertion (A).- Both Assertion (A) and Reason (R) are true but Reason (R) is not a correct explanation of Assertion (A).
- Assertion (A) is true and Reason (R) is false.
- Assertion (A) is false and Reason (R) is true.
How many solid cylinders of radius 6 cm and height 12 cm can be made by melting a solid sphere of radius 18 cm?
Activity: Radius of the sphere, r = 18 cm
For cylinder, radius R = 6 cm, height H = 12 cm
∴ Number of cylinders can be made =`"Volume of the sphere"/square`
`= (4/3 pir^3)/square`
`= (4/3 xx 18 xx 18 xx 18)/square`
= `square`
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 ______.
Four horses are tethered with equal ropes at 4 corners of a square field of side 70 metres so that they just can reach one another. Find the area left ungrazed by the horses.
