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प्रश्न
A bucket open at the top is in the form of a frustum of a cone with a capacity of 12308.8 cm3. The radii of the top and bottom of the circular ends of the bucket are 20 cm and 12 cm respectively. Find the height of the bucket and also the area of the metal sheet used in making it. (Use π = 3.14)
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उत्तर
Let the depth of the bucket is h cm. The radii of the top and bottom circles of the frustum bucket are r1 =20cm and r2 =12cm respectively.
The volume/capacity of the bucket is
`"V" = (1)/(3)π("r"_1^2 + "r"_1"r"_2 + "r"_2^2) xx "h"`
= `(1)/(3)π(20^2 + 20 xx 12 + 12^2) xx "h"`
= `(1)/(3) xx (22)/(7) xx 784 xx "h"`
= `1/3xx22xx112xx"h" "cm"^3`
Given that the capacity of the bucket is 12308.8 Cubic cm. Thus, we have
`(1)/(3) xx 22 xx 112 xx "h" = 12308.8`
⇒ `"h" = (12308.8 xx 3)/(22 xx 112)`
⇒ h = 15
Hence, the height of the bucket is 15 cm
The slant height of the bucket is
`"l" = sqrt(("r"_1 -"r"_2)^2 + "h"^2`
= `sqrt((20 -12)^2 + 15^2)`
= `sqrt(289)`
= 17cm
The surface area of the used metal sheet to make the bucket is
`"S"_1 = π("r"_1 +"r"_2) xx l + π"r"_2^2`
= π x (20 + 12) x 17 + π x 122
= π x 32 x 17 + 144π
= 2160.32cm2
Hence Surface area of the metal = 2160.32cm2
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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?
