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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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