Question

A hemispherical tank full of water is emptied by a pipe at the rate of $3\dfrac{4}{7}$ liters per second. How much time will it take to half empty the tank, if the tank is 3 meters in diameter? (Take $\pi = \dfrac{{22}}{7}$)
A) 16.5 min.
B) 12.8 min.
C) 20.5 min.
D) 18.2 min.

Answer 1

Hint:
In this question, first we will discuss some basic points related to the hemisphere.
A sphere is a set of points in three dimensions and all the points lying on the sphere is equidistant from the center.
In general, we can say that a sphere makes exactly two hemispheres.
As we know that,
The volume of a hemisphere = $\dfrac{2}{3}\pi {r^3}$ cubic units.
Where, $\pi $ is a constant whose value is equal to $\pi = \dfrac{{22}}{7}$ approximately and ‘r’ is the radius of the hemisphere.

Complete step by step solution:
Tank is in the form of hemisphere with Diameter = 3 m
So, Radius = $\dfrac{{Diameter}}{2}$ = $\dfrac{3}{2}$m
Volume of the hemisphere tank = $\dfrac{2}{3}\pi {r^3}$ (Putting values)
$
= \dfrac{2}{3} \times \dfrac{{22}}{7} \times {\left( {\dfrac{3}{2}} \right)^3} \\
 = \dfrac{2}{3} \times \dfrac{{22}}{7} \times \dfrac{3}{2} \times \dfrac{3}{2} \times \dfrac{3}{2} \\
= \dfrac{{99}}{{14}}{m^3} \\
 $
Now, 1m3 = 1000 liters
$\dfrac{{99}}{{14}}{m^3} = \dfrac{{99}}{{44}} \times 1000\,liters$
$ \Rightarrow \dfrac{{99000}}{{14}}liters$
Volume of water to be empty = $\dfrac{1}{2} \times \,$volume of hemispherical tank
$
   \Rightarrow \dfrac{1}{2} \times \dfrac{{99000}}{{14}}\, \\
   \Rightarrow \dfrac{{99000}}{{28}}\,liters \\
 $
Now, it is given that the tank is emptied at $3\dfrac{4}{7}$ liters per second.
Therefore, $\dfrac{{25}}{7}\,$liters per second
Time taken to empty $\dfrac{{25}}{7}\,$liters = 1 second
Time taken to empty 1 liter = $\dfrac{1}{{\dfrac{{25}}{7}}}$second = $\dfrac{7}{{25}}$second
Time taken to empty $\dfrac{{99000}}{{28}}$liters = $\dfrac{7}{{25}}$ X $\dfrac{{99000}}{{28}}$ = 990 second (solving this equation)
We get, 990 second (converting seconds into minutes)
$ \Rightarrow \dfrac{{990}}{{60}}$minutes = 16.5 minutes
Hence, Time taken to half empty the tank is 16.5 minutes.

The correct option is A.

Note:
Students can also solve this question in a short way. Let’s see how we can solve it.
Given,
Diameter = 3 meter
Radius = 1.5 meter ($\dfrac{3}{2} = 1.5$meter)
Volume of hemisphere = $\dfrac{2}{3}\pi {r^3}$
We get,$
   = \dfrac{2}{3} \times \,3.14\, \times \,{(1.5)^3} \\
   = \dfrac{2}{3} \times \,3.14\, \times \,1.5 \times \,1.5 \times \,1.5 \\
   = 7.065\,\,{m^3} \\
   = 7065\,l\,\,(1{m^3}\, = 1000l) \\
 $
Empty Rate = $\dfrac{{25}}{7}\,$per second
Now, let time taken to empty half volume of the tank = ‘t’
Then according to the question, we get,
$ \Rightarrow \dfrac{{25}}{7} \times t = \dfrac{1}{2} \times volume$(volume = 7065L)
$
   \Rightarrow t = \dfrac{1}{2} \times 7065 \times \dfrac{7}{{25}} \\
   \Rightarrow t = \dfrac{{9891}}{{10}}\sec \\
   \Rightarrow t = 989.1\sec \\
 $
Now converting the above seconds in minute:
$ \Rightarrow \dfrac{{989.1}}{{60}}$
$ \Rightarrow \dfrac{{9891}}{{600}}$
$ \Rightarrow $16.435
$ \Rightarrow $16.5 minutes (Answer)