Question

Water flows at the rate of 10 m per min from cylindrical pipe 5mm in diameter. How long will it take to fill up a conical vessel whose diameter at the base is 40cm and depth 24cm ?
(a)48min 15sec
(b)51min 12sec
(c)52min 1sec
(d)55min

Answer 1

Hint: We use the formula of volume of a cone which is \[V=\dfrac{1}{3}\pi {{r}^{2}}h\], where r is the radius of the base of the cone and h is the height/depth of the cone and the volume of the water flowing per minute to find the answer.

Complete step-by-step answer:
Given that water flows at the rate of 10m/min of the cylindrical pipe.
Let the radius of the cylindrical pipe be R.
Given the diameter of the cylindrical pipe is 5mm, therefore the Radius of the pipe R will be
\[R=\dfrac{5}{2}=2.5mm=0.25cm\]
Speed of water = 10 m/min, because all our calculation is to be done in cm so we convert this value in cm, then the Speed of water becomes 1000 cm/min, which ultimately becomes the height of the cylindrical pipe, let it be H as shown in the figure below.

Then H=1000cm/min
Because the pipe is cylindrical then the Volume of water that flows in 1 minute is calculated by the Volume of the cylinder which is
\[V'=\pi {{R}^{2}}H\]
So, after substituting the values we get
\[\begin{align}
& V'=\pi {{(0.25)}^{2}}(1000) \\
& \\
& \Rightarrow V'=\pi (62.5) \\
& \\
& \Rightarrow V'=\dfrac{22}{7}(62.5) \\
& \\
& \Rightarrow V'=196.4285c{{m}^{3}}...........(i) \\
\end{align}\]

Considering the calculations for the cone now,
Given, the diameter of the cone is 40cm then radius of the cone will be \[\dfrac{d}{2}=\dfrac{40}{2}=20cm\], let it be r
That is r=20cm and the Height of the cone is h= 24cm
Now we put these values in the formula of Volume of the cone to get the capacity of the conical vessel
We have \[V=\dfrac{1}{3}\pi {{r}^{2}}h\]
\[\begin{align}
& \Rightarrow V=\dfrac{1}{3}\left( \dfrac{22}{7} \right)\left( {{20}^{2}} \right)\left( 24 \right) \\
& \\
& \Rightarrow V=\dfrac{1}{3}\left( \dfrac{22}{7} \right)\left( 400 \right)\left( 24 \right) \\
& \\
& \Rightarrow V=\dfrac{1}{3}\left( \dfrac{22}{7} \right)\left( 9600 \right) \\
& \\
& \Rightarrow V=\dfrac{1}{3}\left( 30171.42 \right) \\
& \\
& \Rightarrow V=10057.14 \\
\end{align}\]
Hence, \[V=10057.14c{{m}^{3}}.........(ii)\]
Now we use equation (i) and (ii) to get the result
Time required to fill the vessel = \[\dfrac{capacity\text{ }of\text{ }the\text{ }conical\text{ }vessel}{volume\text{ }of\text{ }water\text{ }flowing\text{ }per\text{ }minute}\]
That is Time required to fill the vessel = \[\dfrac{equation(ii)}{equation(i)}\]
Time required to fill the vessel =\[\dfrac{10057.14}{196.42}\]
Time required to fill the vessel =51.27min
Hence, the time taken to fill the conical vessel is 51mins and 12 secs i.e. option (b).

Note:A minor but very important error in this question could be not changing the units of every term of the question to cm or to mm. We always convert all the given terms in one single unit and not multiple ones.