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Question
A cylindrical bucket of height 32 cm and base radius 18 cm is filled with sand. This bucket is emptied on the ground and a conical heap of sand is formed. If the height of the conical heap is 24 cm, find the radius and slant height of the heap.
Solution
Given, radius of the base of the bucket = 18 cm
Height of the bucket = 32 cm
So, volume of the sand in cylindrical bucket
= πr2h
= π(18)2 × 32
= 10368π
Also, given height of the conical heap (h) = 24 cm
Let radius of heap be r cm.
Then, volume of the sand in the heap
= `1/3 pir^2h`
= `1/3 pir^2 xx 24`
= 8πr2
According to the question,
Volume of the sand in cylindrical bucket = Volume of the sand in conical heap
⇒ 10368π = 8πr2
⇒ 10368 = 8r2
⇒ r2 = `10368/8` = 1296
⇒ r = 36 cm
Again, let the slant height of the conical heap = l
Now, l2 = h2 + r2
= (24)2 + (36)2
= 576 + 1296
= 1872
∴ l = 43.267 cm
Hence, radius of conical heap of sand = 36 cm
And slant height of conical heap = 43.267 cm
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