The height and base diameter of a solid cube are 20 cm and 20 cm. One cube of max. possible size is

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29 Jul 2016 Volume of Tower

The height and base diameter of a solid cube are 20 cm and 20 cm. One cube of max. possible size is cut from the cone touching the base, after this second cone is cut above it from the cone of max possible size, the process is repeated infinite times, all cubes are one above the other and seems like a tower having largest cube at bottom and smallest at top. Calculate the volume of tower.

OPtion

1) 1271.42
2) 1285.71
3) 1157.14
4) 1244.58
5) 1286.86
6) 1171.42
7) 1189.65
8) 1142.85
9) 1185.71
10)1215.42

Solution

Let dimensions of largest cube are x cm then by symmetry
20/10 = x/ (10- x/2)
=> 20-x = x
=> x = 10 cm
After cutting this cube, remaining cone has height and diameter as 10 cm, just half of initial so dimensions of second cube will be half of first similarly others so net volume of tower
= 10^3 + (10/2)^3 + (10/4)^3 + ........... infinite
= 1000(1 + 1/8 + 1/8^2 + ................. infinite
= 1000(1/ (1- 1/8))
= 1000*8/7 = 8000/7 cm^3
1142.85 cm^3
Option 8)

Correct Option: 8 Best Solution (2)

G.S.Kantharao   8 years AGO

Measurements of first cone is d = 20cm, r =10cm and h =20cm.
Side of a maximum possible cube is = 10cm
Side of next maximum cube is = 5cm
Next ones are 2.5cm, 1.25cm etc
Sum of their volumes are = 10^3 + 5^3 + (2.5)^3 + (1.25)^3 + .........
=10^3 + (10/2)^3 +(10/4)^3 +..........
=10^3{1 + 1/(2^3) +1/(4^3) +......}
= 10^3{1/(1-1/8)}
= 1000 x 1/(7/8)
= 1000 x 8/7
= 1000 x 1.14285
=1142.85
Hence right choice is 8

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Aakash Beniwal   8 years AGO

size of any cube inside cone with equal diameter and height is = 1/2 of diameter.
hence cube volume= 10^3 cm= 1000
now remaining cone is of diameter 10 cm.
so volume of cube= 5^3 = 125 cm^3
similarly next cone with 5 cm diameter and cube with 2.5 cm side.
hence volume = 2.5 ^ 3 cm ^3 ....
and so on... 1000 + 125 + 15.625 + 1.95 + ... = 1142.8 (8th option)
Or can use
summation of (20/2k)^3 from k=1 to infinity....

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