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Question
From a cubical piece of wood of side 21 cm, a hemisphere is carved out in such a way that the diameter of the hemisphere is equal to the side of the cubical piece. Find the surface area and volume of the remaining piece.
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Solution
We have,
the edge of the cubical piece, a = 21 cm and
the radius of the hemisphere, `r = a/2=21/2`
The surface area of the remaining piece = TSA of cube + CSA of hemisphere -Area of
= 6a2 + 2πr2 - πr2
= 6a2 + πr2
`= 6 xx 21xx21xx22/7xx21/5xx21/2`
`= 21xx21(6 + 22/(7+4))`
`= 21xx21(6+11/14)`
`=21xx21((84+11)/14)`
`= 21xx3(95/2)`
= 2992.5 cm2
Also,
Volume of the remaining piece = volume of the cube - volume of the hemisphere
`= a^3 - 2/3pir^3`
`= 21xx21xx21 - 2/3xx22/7xx(21/2)xx(21/2)xx(21/2)`
`= 21xx21xx21xx(1-2/3xx22/7xx1/2xx1/2xx1/2)`
`= 21xx21xx21(1/1 - 11/42)`
`=21xx21xx21xx(42-11)/42`
`= 21xx21xx(31/2)`
= 6835.5 cm3
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Khurja is a city in the Indian state of Uttar Pradesh famous for the pottery. Khurja pottery is traditional Indian pottery work which has attracted Indians as well as foreigners with a variety of tea sets, crockery and ceramic tile works. A huge portion of the ceramics used in the country is supplied by Khurja and is also referred as "The Ceramic Town". One of the private schools of Bulandshahr organised an Educational Tour for class 10 students to Khurja. Students were very excited about the trip. Following are the few pottery objects of Khurja.
Students found the shapes of the objects very interesting and they could easily relate them with mathematical shapes viz sphere, hemisphere, cylinder etc. |
Maths teacher who was accompanying the students asked the following questions:
- The internal radius of hemispherical bowl (filled completely with water) in I is 9 cm and the radius and height of the cylindrical jar in II are 1.5 cm and 4 cm respectively. If the hemispherical bowl is to be emptied in cylindrical jars, then how many cylindrical jars are required?
- If in the cylindrical jar full of water, a conical funnel of the same height and same diameter is immersed, then how much water will flow out of the jar?
A tent is in the shape of a cylinder surmounted by a conical top. If the height and radius of the cylindrical part are 3 m and 14 m respectively, and the total height of the tent is 13.5 m, find the area of the canvas required for making the tent, keeping a provision of 26 m2 of canvas for stitching and wastage. Also, find the cost of the canvas to be purchased at the rate of ₹ 500 per m2.
