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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.
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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.
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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?
Statement A (Assertion): Total Surface area of the top is the sum of the curved surface area of the hemisphere and the curved surface area of the cone.
Statement R( Reason): Top is obtained by joining the plane surfaces of the hemisphere and cone together.
