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Question
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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?
Solution
a. Given, the radius by the hemispherical bowl, r1 = 9 cm
Radius of the cylindrical jar, r2 = 1.5 cm
Height of cylindrical jar, h2 = 4 cm
Now, Volume of hemispherical bowl = `2/3 πr_1^3`
= `2/3 π(9)^3`
And Volume of cylindrical jar = `πr_2^2h_2`
= π(1.5)2 × 4
Required number of cylindrical jar = `"Volume of hemispherical bowl"/"Volume of cylindrical jar"`
= `(2/3 π(9)^3)/(π(1.5)^2 xx 4)`
= `(2 xx 9 xx 9 xx 9)/(3 xx 1.5 xx 1.5 xx 4)`
= `(3 xx 9 xx 9 xx 10 xx 10)/(15 xx 15 xx 2)`
= `(24,300)/450`
= 54
Hence, 54 cylindrical jars are required.
b. Volume of water flow out of the jar = Volume of the conical funnel
= `1/3 πr_2^2h_2`
= `1/3 xx22/7 xx (1.5)^2 xx 4`
= `1/3 xx 22/7 xx 1.5 xx 1.5 xx 4`
= `(22 xx 15 xx 15 xx 4)/(3 xx 7 xx 10 xx 10)`
= `19800/2100`
= 9.43 cubic cm
Therefore, the water flowing out of the jar is 9.43 cubic cm.
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