Question

A hemispherical tank, full of water, is emptied by a pipe at a rate of 25/7 liters per second. How much time will it take to empty half the tank if the diameter of the base of the tank is 3m?

Answer 1

Hint – In this question find the volume of the hemisphere using the direct formula for volume that is $\dfrac{2}{3}\pi {r^3}$, then calculate half of this volume. Use a unitary method to find the time in which this half volume can be emptied as rate is 25/7 liters per second.
Complete step-by-step solution -

As we know that the volume (V) of the hemisphere is $\dfrac{2}{3}\pi {r^3}$cubic units.
Where r is the radius of the hemisphere.
Now it is given a hemispherical bowl full of water with diameter 3 meter.
So radius (r) is half of the diameter.
Therefore, r = (d/2) = (3/2) = 1.5 meter.
So the volume of the hemisphere is,
$ \Rightarrow V = \dfrac{2}{3}\pi {r^3} = \dfrac{2}{3} \times \dfrac{{22}}{7} \times {\left( {1.5} \right)^3} = \dfrac{{148.5}}{{21}}$ cubic meters.
Now as we know that $1{m^3}$ = 1000 liters.
Therefore $\dfrac{{148.5}}{{21}}{\text{ }}{{\text{m}}^3} = \dfrac{{148500}}{{21}}$ liters.
So the total volume of water in a hemispherical tank is (V) = $\dfrac{{148500}}{{21}}$ liters.
So the volume of half empty tank = $\dfrac{V}{2} = \dfrac{1}{2} \times \dfrac{{148500}}{{21}} = \dfrac{{74250}}{{21}}$ liters.
Now it is given the time required to empty the tank by (25/7) liters of water in 1 sec.
So the time required to empty the tank by 1 liter of water = $\dfrac{1}{{\dfrac{{25}}{7}}} = \dfrac{7}{{25}}$ seconds.
So the time required to empty the tank by $\dfrac{{74250}}{{21}}$ liters of water = $\dfrac{7}{{25}} \times \dfrac{{74250}}{{21}} = 990$ seconds.
Now as we know that 60 seconds = 1 minute.
Therefore 1 second = $\dfrac{1}{60}$ minutes.
Therefore 990 seconds = $\dfrac{990}{60}$ = 16.5 minutes.
So the time required to empty half the tank is 16.5 minutes.
So this is the required answer.

Note – The key steps in this problem was the unit conversion of ${m^3} \leftrightarrow {\text{litres}}$ and time in seconds to minutes. We need to remember some basic units like $1{m^3}$ = 1000 liters, this conversion is important as meter is a unit of measurement of length whereas liters is of quantity. 1 minute consists of 60 seconds and 1 hour consists of 60 minutes.