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प्रश्न
Match the following columns:
| Column I | Column II |
| (a) The radii of the circular ends of a bucket, in the form of the frustum of a cone of height 30 cm, are 20 cm and 10 cm respectively. The capacity of the bucket is ........cm3. |
(p) 2418π |
| (b) The radii of the circular ends of a conical bucket of height 15 cm are 20 and 12 cm respectively. The slant height of the bucket is ........ cm. |
(q) 22000 |
| (c) The radii of the circular ends of a solid frustum of a cone are 33 cm and 27 cm and its slant height is 10 cm. The total surface area of the bucket is .........cm2. |
(r) 12 |
| (d) Three solid metallic spheres of radii 3 cm, 4 cm and 5 cm are melted to form a single solid sphere. The diameter of the resulting sphere is ........ cm. |
(s) 17 |
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उत्तर
(a)
Let R and r be the top and base of the bucket and let h be its height.
Then, R = 20 cm, r = 10 cm and h = 30 cm.
Capacity of the bucket = Volume of the frustum of the cone
`= (pi"h")/3("R"^2 + "r"^2 + "Rr")`
`= 22/7xx1/3xx30xx[(20)^2 + (10^2) + (20xx10)] "cm"^3`
`= 22/7xx[400+100+200]"cm"^3`
`=(220/7xx700)"cm"^3`
= 22000 cm3
Hence, (a) ⇒ (q)
(b)
Let R and r be the top and base of the bucket and let h be its height.
Then, R = 20 cm, r = 12 cm and h = 15 cm
`"Slant height of the bucket" "l" = sqrt(h^2 + ("R"-r)^2) `
`=sqrt((15)^2 + (20-12)^2)`
`=sqrt(225+64)`
`=sqrt(289) `
= 17 cm
Hence, (b) ⇒ (s)
(c)
Let R and r be the top and base of the bucket and let be its slant height.
Then, R = 33 cm, r = 27 cm and h = 10 cm
Total surface area of the bucket `= pi ["R"^2 +"r"^2+"l"("R" + r)]`
`= pixx[(33)^2 + (27)^2 + 10xx(33+27)]`
`= pi xx [1089 + 729 + 600]`
`= 2418pi "cm"^2`
Hence, (c) ⇒ (p)
(d)
Let the diameter of the required sphere be d.
Then, volume of the sphere`=4/3 pi"r"^3`
`= 4/3pi("d"/2)^3`
Therefore,
`4/3pi("d"/2)^3 = 4/3pi(3)^3 + 4/3pi(4)^3 + 4/3pi(5)^3`
`=> 4/3pi"d"^3/8 = 4/3pixx[(3)^3+(4)^3 + (5)^3`
`= "d"^3/8 = 216`
⇒ d3 = 1728
⇒ d3 = 123
⇒ d = 12 cm
Hence, (d) ⇒ (r)
| Column I | Column II |
| (a) The radii of the circular ends of a bucket, in the form of the frustum of a cone of height 30 cm, are 20 cm and 10 cm respectively. The capacity of the bucket is ........cm3. |
(q) 22000 |
| (b) The radii of the circular ends of a conical bucket of height 15 cm are 20 and 12 cm respectively. The slant height of the bucket is ........ cm. |
((s) 17 |
| (c) The radii of the circular ends of a solid frustum of a cone are 33 cm and 27 cm and its slant height is 10 cm. The total surface area of the bucket is .........cm2. |
(p) 2418π |
| (d) Three solid metallic spheres of radii 3 cm, 4 cm and 5 cm are melted to form a single solid sphere. The diameter of the resulting sphere is ........ cm. |
(r) 12 |
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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?
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