Question

Water flows at the rate of 10 m per minute through a cylindrical pipe of 5 mm in
diameter. How long would it take to fill a conical vessel whose diameter of the base
is 40 cm and depth 24 cm?
​

Answer 1

Explanation -

Given that - Water flows at the rate of 10 m per minute through a cylindrical pipe of 5 mm in diameter. Conical vessel's diameter of the base is 40 cm and it's depth is 24 cm.

⠀⠀⠀⠀⠀⠀⠀⠀Means,

• In one minute the length of he water column in the cylindrical pipe = 10 metres

• Diameter of the pipe = 5 mm

• Diameter of the base of the conical vessel = 40 cm

• And the depth of the conical vessel = 24 cm

To find - How long would it take to fill a conical vessel.

Solution - It take 51.2 minutes to fill a conical vessel.

Using concepts -

• Formula to covert diameter into radius.

⠀⠀⠀⠀↦Radius = Diameter/2

• Formula to covert metres into centimetres.

⠀⠀⠀↦1 metre = 100 centimetres

• Formula to covert millilitres into centimetres.

⠀⠀⠀↦1 mm = 1/10 centimetres

• One minute the length of he water column in the cylindrical pipe, formula (according to the question),

⠀↦Volume = πr² × 10m (when converted)

• Formula to find volume of cone(vessel's shape) (according to the question),

⠀⠀⠀⠀↦Volume = 1/3 πr² × depth

• Formula to find that how long would it take to fill a conical vessel(according to the question),

⠀⠀⠀↦Formula to find volume of cone/One minute the length of he water column in the cylindrical pipe formula.

When all the units are covered, we get the following results -

• In one minute the length of he water column in the cylindrical pipe = 10 metres i.e., 1000 centimetres

• Diameter of the pipe = 5 mm i.e., 5/10 centimetres means, 1/2 centimetres

∴ Radius = 1/2/2 = 1/4 centimetres

• Diameter of the base of the conical vessel = 40 cm

∴ Radius = 40/2 = 20 centimetres

Now according to the question,

~ One minute the length of he water column in the cylindrical pipe, formula (according to the question),

⟹ Volume = πr² × 10m

⠀⠀⠀⠀⠀⠀⠀⠀Means,

⟹ Volume = πr² × 1000

⟹ Volume = π × 1/4 × 1/4 × 1000

~ Formula to find volume of cone(vessel's shape) (according to the question),

⟹ Volume = 1/3 πr² × depth

⟹ Volume = 1/3 × π × 20 × 20 × 24

~ Formula to find that how long would it take to fill a conical vessel(according to the question),

⟹ Formula to find volume of cone/One minute the length of he water column in the cylindrical pipe formula.

${\sf{\implies \dfrac{20 \times 20 \times 24}{3} \times \dfrac{4 \times 4}{1000}}}$

${\sf{\implies \dfrac{400 \times 24}{3} \times \dfrac{16}{1000}}}$

${\sf{\implies \dfrac{9600}{3} \times \dfrac{8}{500}}}$

${\sf{\implies \dfrac{9600}{3} \times \dfrac{4}{250}}}$

${\sf{\implies \dfrac{9600}{3} \times \dfrac{2}{125}}}$

${\sf{\implies 3200 \times \dfrac{2}{125}}}$

${\sf{\implies 51.2 \: minutes}}$

Answer 2

Solution:

Amount of water required to fill the conical vessel

= volume of conical vessel

=  1/3     π(20)²  ×24=3200πcu.cm ___ (1)

Amount of water that flows out of cylindrical

pipe in 1 minute = π×(5/20)²         ×10×100  

                          = 62.5\pi cu.cm .... (2)

From (1) & (2)

Time required to fill the vessel =      3200π/62.5π

 = 51.2minutes.

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