Question

Water flows out through a circular pipe whose internal diameter is 2cm, at the rate of 6 meter per second into a cylindrical tank, the radius of whose base is 60cm. By how much will the level of water rise in 30 minutes?

Answer 1

Hint: In this question the height of the cylindrical tank is to be taken out at time 30 minutes. Use the direct formula for volume of cylindrical tank which is $\pi {r^2}h$, but the volume of water eventually following into the tank through the pipe should be equal to this volume (obtained through direct formula). Use this concept to get the height.

Complete Step-by-Step solution:
The internal diameter of the circular pipe is = 2 cm.
So the radius (r) of the circular pipe is half of the diameter = 1 cm.
Now it is given that the water flows out at the rate of 6 m/s into the cylinder.
As we know 1 m = 100 cm.
Therefore water flows out at the rate of 600 cm/sec.
So the volume (V) of the water that flows out in 1 sec is the multiplication of the area of the circle and the rate of water at which it flows.
Now as we know that the area of the circle is $\pi {r^2}$ where r is the radius of the circle.
$ \Rightarrow V = \pi {r^2} \times {\text{rate of water it flows}}$
$ \Rightarrow V = \pi \times {1^2} \times 600$
$ \Rightarrow V = 600\pi {\text{ c}}{{\text{m}}^3}$.
So the volume of water that flows in 30 minutes (i.e. $30 \times 60$ seconds) is
$ \Rightarrow V = 600\pi \left( {30 \times 60} \right){\text{ c}}{{\text{m}}^3}$……………………… (1)
Now this volume of water must be equal to the volume of the cylinder in which it flows.
Now as we know that the volume of cylinder (VC) is $\pi {r^2}h$ where r is the base radius of the cylinder and h is the height of the cylinder.
Let us assume that h = level of water rises in the cylinder and the base radius of the cylinder is given which is 60 cm.
$ \Rightarrow {V_C} = \pi {\left( {60} \right)^2}h{\text{ c}}{{\text{m}}^3}$………………………………………… (2)
Now equate equation (1) and (2) we have,
$ \Rightarrow 600\pi \left( {30 \times 60} \right) = \pi {\left( {60} \right)^2}h$
Now simplify the above equation we have,
$ \Rightarrow 10\left( {30} \right) = h{\text{ }}$
$ \Rightarrow h = 300$ cm.
Now 100 cm = 1 m
Therefore 300 cm = 3 m.
$ \Rightarrow h = 3$ m.
So level of water rise in the cylinder is 3 meter,
Hence option (B) is correct.

Note: The water which flows through the pipes into the tank has the volume as area of circle multiplied with the rate flow of water. The pipe is given to be circular in shape thus this volume can be calculated. Some students must be wondering that the circle is a 2-d shape so how to take out it’s volume. We are not taking out the volume of the circle instead we are taking out the volume of water flowing through this circular section.