Water is flowing at the rate of 15 km/h through a pipe of diameter

Q) Water is flowing at the rate of 15 km/h through a pipe of diameter 14 cm into a cuboidal pond which is 50 m long and 44 m wide. In what time will the level of water in pond rise by 21 cm? What should be the speed of water if the rise in water level is to be attained in 1 hour?

Ans:

Here, in this question, we will consider a pipe as a cylinder with pipe’s circular opening as cylinder’s circular base and speed of water to be taken as cylinder’s height in meter per hour terms.

We know that the volume of a cylinder is given by: $\pi$ r²h

Here, we are given that the diameter of the cylinder’s base is 14 cm.

Therefore, radius of the cylinder’s base is $\frac{14}{2}$ = 7 cm = $\frac{7}{100}$ m

Since we will take speed of the water as height of the cylinder,

Therefore, height of the cylinder = 15 km per hour = 15,000 m/hour (given)

Hence, the volume of the water coming from the pipe per hour:

= $(\frac{22}{7}) (\frac{7}{100})^2}$ x 15000 = 231 m³per hour …………… (i)

A) Time taken to rise the water level by 21cm in the pond:

For the water level be risen by 21 cm, we need to calculate the the volume of the water filled in the pond. We know that the volume of a cuboid is given by: Length x width x height

Therefore, water filled in the pond till height of 21 cm (or $\frac{21}{100}$ m):

= 50 x 44 x $\frac{21}{100}$ = 462 m³………. (ii)

Hence, the time taken to fill this volume = $\frac{volume~of~water~in~the~pond}{volume~of~the~water~coming~from~the~pipe~per~hour}$

Substituting values from equations (i) and (ii), we get

Time taken to fill the volume = $\frac{462}{231}}$ = 2 hours

B) Speed of water required to fill the pond till desired height in 1 hour:

Here, the time taken to fill the pond will be calculated by:

Time taken = $\frac{volume~of~water~in~the~pond}{volume~of~the~water~coming~from~the~pipe~per~hour}$

Since, Volume of the water coming from pipe per hour = area of the circular opening of the pipe x speed of the water coming from the pipe per hour

$\therefore$ The time taken to fill the pond = $\frac{volume~of~water~in~the~pond}{(area~of~the~circular~opening~of~the~pipe) (speed~of~the~water~coming~from~the~pipe~per~hour)}$

We can clearly see that the here time taken is inversely proportional to the speed of water from pipe. Lesser is the speed of water in the pond, more will be the time to fill and higher the speed of water, less will be the time taken.

Hence, to reduce the time to 1 hour from 2 hours, speed of water needs to be doubled.

Since initially, speed of water was 15 km per hour and it took 2 hours to get the water filled till21 cm height.

Hence, now to get the water filled at same height, but in half of the time, speed of the water will be 2 times i.e. 2 x 15 = 30 km per hour.

Alternate solution:

We have just calculated in part A that to get the water level risen by 21 cm in 1 hour, we need to get the 462 m³ volume of the water inside the pond [refer equation (i) ]

Now, in this question, this volume of water is desired to be filled by the water coming from the pipe in 1 hour.

$\therefore$ , Volume of water coming from pipe in one hour = 462 m³ ………. (iii)

Since the volume of the water from a circular pipe is given by (area of the circular opening of the pipe) x (speed of the water coming from the pipe per hour)………. (iv)

Here, area of the circular opening of the pipe = $\pi$ r ²

= $(\frac{22}{7})(\frac{7}{100})^2 = \frac{77}{5000}$ m²…….. (v)