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A cylindrical pipe with a diameter of 5 mm is filled with water at a speed of 10 m per minute. How long would it take to fill a conical vessel that was 24 cm deep and 40 cm in diameter at the surface?

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50.2 min

51.2 min

50.4 min

51.4 min

answer is B.

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Detailed Solution

It is given that a cylindrical pipe with a diameter of 5 mm is filled with water at a speed of 10 m per minute.
We know that the volume of the cone is

13 π r 2 h,

where r is the radius of the base and h is the cone's height.
The volume of a cylinder is

π r 2 h,

where r is the radius of the base and h is the cylinder's height.
It is given that pipe’s diameter

= 5 mm = 0.5 cm .

Therefore, the pipe’s radius

= 0.5 2 = 0.25 cm .

Length of the pipe

= 10 m = 1000 cm .

Calculate the volume that flows in 1 min as shown,

π × 0.25 2 × 1000 cm 3

Calculate the volume of the conical vessel as shown,

13 π × 202 × 24 cm 3

Calculate the desired time as shown,

∴ T = 13 π × 202 × 24 π × 0.25 2 × 1000 min ⇒ T = 13 π × 400 × 24 π × 0.0625 × 1000 min ⇒ T = 51.2 min ⇒ T = 51 min 12 sec

Therefore, the time needed to fill a conical vessel is 51.2 min.
Hence, option (2) is correct.

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