Question

A hemispherical tank full of water is emptied by a pipe at the rate of $ 3\dfrac{4}{7} $ litres per second. How much time will it take to empty half the tank, if it is $ 3m $ in diameter. (Take $ \pi = \dfrac{{22}}{7} $ )

Answer 1

Hint: First we will use the standard formula to find volume. As per the requirement convert measure of diameter into radius and then convert the volume into litres and then find the time taken with the help of the reference speed or rate of the flow of the water.

Complete step-by-step answer:
Given that the diameter of the tank is $ d = 3m $
We know that radius is always half of the diameter.
 $ \Rightarrow r = \dfrac{d}{2} = \dfrac{3}{2}m $
We know that the volume of the hemispherical tank can be given by –
 $ V = \dfrac{2}{3}\pi {r^3} $
Place values in the above equation –
 $ V = \dfrac{2}{3} \times \left( {\dfrac{{22}}{7}} \right) \times {\left( {\dfrac{3}{2}} \right)^3} $
Simplify the above equation –
 $ V = \dfrac{2}{3} \times \left( {\dfrac{{22}}{7}} \right) \times \left( {\dfrac{3}{2}} \right) \times \left( {\dfrac{3}{2}} \right) \times \left( {\dfrac{3}{2}} \right) $
Common factors from both the denominator and the numerator cancel each other, so remove
 $ \Rightarrow V = \dfrac{{99}}{{14}}{m^3} $ .... (A)
Convert cubic metre in litres.
 $ 1{m^3} = 1000{\text{ }}litres $
Use equation (A) and convert it in litres-
 $
   \Rightarrow V = \dfrac{{99}}{{14}} \times 1000{\text{ litres}} \\
   \Rightarrow V = \dfrac{{99000}}{{14}}litres \\
  $
So capacity of the tank is $ \dfrac{{99000}}{{14}}{\text{ litres}} $ , and given that tank is to be emptied half.
Therefore, divide the total capacity by two.
 $ \Rightarrow \dfrac{{99000}}{{14 \times 2}}{\text{ litres = }}\dfrac{{99000}}{{28}}{\text{ litres}} $
Convert the given integer $ 3\dfrac{4}{7} $ in fraction.
 $ 3\dfrac{4}{7} = \dfrac{{21 + 4}}{7} = \dfrac{{25}}{7} $
Water is emptied by a pipe at the rate of $ 3\dfrac{4}{7} $ litres per second
 $ \Rightarrow \dfrac{{25}}{7}{\text{ Litres = 1 Sec}} $
Therefore, $ \dfrac{{99000}}{{28}}{\text{ litres = ?}} $
Do- cross multiplication and simplify-
Time taken $ = \dfrac{{99000}}{{28}} \times \dfrac{7}{{25}}\sec $
Simplify the above equation –
Time taken $ = \dfrac{{693000}}{{700}}\sec $
Take out common factors and simplify
Time taken $ = 990\operatorname{Sec} $
Convert the time taken in seconds into minutes
Time taken $ = \dfrac{{990}}{{60}}\operatorname{Sec} $
 Time taken $ = 16.5\min $
Hence, time taken to empty half the tank is $ 16.5 $ minutes.

Note: The most important here is the standard formula and its simplification. Always remember the basic conversion relations and convert as per the need. Remember the difference among the units such as one second and one minute both gives us the time.