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प्रश्न
504 cones, each of diameter 3.5 cm and height 3 cm, are melted and recast into a metallic sphere. Find the diameter of the sphere and hence find its surface area.
[Use π=22/7]
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उत्तर
No. of cones = 504
Diameter of a cone = 3.5 cm
Radius of the cone, r = 1.75 cm
Height of the cone, h = 3 cm
Volume of a cone
`=1/3pir^2h`
`=1/3xxpixx(3.5/2)^2xx3`
`=1/3xxpixx3.5/2xx3.5/2xx3 " cm"^2`
Volume of 504 cones
`=504xx1/3xxpixx3.5/2xx3.5/2xx3 " cm"^2`
Let the radius of the new sphere be ‘R’.
Volume of the sphere `=4/3 pir^3 `
Volume of 504 cones = Volume of the sphere
`504xx1/3xxpixx3.5/2xx3.5/2xx3=4/3 piR^3`
`(504xx1xxpixx3.5xx3.5xx3xx3)/(3xx2xx2xx4xxpi)=R^3`
`R^3=(504xx3xx49)/64`
`R^3=(8xx27xx7^3)/64`
`R=(2xx3xx7)/4`
`R=21/2=10.5 cm`
Radius of the new sphere = 10.5 cm
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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?
