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प्रश्न
The perimeters of the ends of a frustum of a right circular cone are 44 cm and 33 cm. If the height of the frustum be 16 cm, find its volume, the slant surface and the total surface.
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उत्तर
The height of the frustum of the cone is h= 16 cm. The perimeters of the circular ends are 44 cm and 33 cm. Let the radii of the bottom and top circles are r1 cm and r2 cm respectively. Then, we have
2πr1= 44
⇒ πr1= 22
⇒ `r_1+(22xx7)/22`
⇒ r1 = 7
2πr2 = 33
⇒ `pi r_2=33/2`
⇒ `r_2=33/2xx7/22`
⇒ `r_2=21/4`
The slant height of the bucket is
`l=sqrt((r_1-r_2)^2+h^2)`
`=sqrt((7_21/4)^2+h^2)`
= 16.1 cm
The curved/slant surface area of the frustum cone is
`= pi(r_1+r_2)xxl`
`=(pir_1+pir_2)xxl`
= (22 + 16.5)xx16.1
= 619.85 cm2
Hence Curved surface area = 619.85cm2
The volume of the frustum of the cone is
`V=1/3pi(r_1^2+r_1r_2+r_2^2)xxh`
`=1/3pi(7^2+7xx5.25+5.25^2)xx16`
= 1900 cm3
Hence Volume of frustum = 1900cm3
The total surface area of the frustum cone is
`=pi(r_1+r_2)xxl+pi r_1^2+pi r_2^2`
`=619.85+22/7xx7^2+22/7xx5.25^2`
= 860.25 Square cm
Hence Total surface area = 860.25 cm2
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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?
The ratio of total surface area of a solid hemisphere to the square of its radius is ______.
