Question

Inner diameter of a hemispherical tank is 3m.
a)How many litres of water does it hold?
b)In what time will the tank fill if water flows at a speed of 1/2 metre per second through a pipe of diameter 2 cm?​

Answer 1

Answer:

given d = 3m => r = 1.5 m

V(hemisphere) = 2/3*pi*r³

=> 2/3*22/7*1.5*1.5*1.5

=> 7. 07 cubic metres

=> 7.07 × 1000 litres [ 1 cu m = 1000 litres ]

=> 7,070 litres

CSA of pipe = pi*r² = 22/7 *1² = 22/7 sq cm

=> 0.000314 sq m

water flowing through this pipe in one second

= 0.000314*0.5 = 0.000157 cu m/sec

=> 0.000157×1000 = 0.157 litres/sec

=> therefore time taken to fill the hemispherical tank of 7,070 litres = 7070÷0.157

=> 45,032 seconds = 45032÷3600 = 12.5 hours

it takes 12.5 hours to fill the tank

Answer 2

The inner radius of the hemispherical tank is $\sf{r=\dfrac{3}{2}\ m.}$

The volume of the hemispherical tank is,

$\longrightarrow\sf{V=\dfrac{2}{3}\,\pi r^3}$

$\longrightarrow\sf{V=\dfrac{2}{3}\,\pi\left(\dfrac{3}{2}\right)^3}$

$\longrightarrow\sf{V=7.069\ m^3}$

Since $\sf{1\ m^3=1000\ L,}$

$\longrightarrow\sf{\underline{\underline{V=7069\ L}}}$

Hence the tank can hold $\bf{7069\ L}$ of water.

Here water flows through the pipe of radius $\sf{r'=1\ cm=0.01\ m}$ at a speed of $\sf{v=\dfrac{1}{2}\ m\,s^{-1},}$ where the pipe is assumed to be cylindrical in shape.

The volume of water flowing through the pipe per unit time is,

$\longrightarrow\sf{V'=\pi(r')^2v}$

$\longrightarrow\sf{V'=\pi(0.01)^2\times\dfrac{1}{2}}$

$\longrightarrow\sf{V'=1.571\times10^{-4}\ m^3\ s^{-1}}$

$\longrightarrow\sf{V'=0.1571\ L\ s^{-1}}$

Hence time taken to fill the tank is,

$\longrightarrow\sf{t=\dfrac{V}{V'}}$

$\longrightarrow\sf{t=\dfrac{7069}{0.1571}}$

$\longrightarrow\sf{\underline{\underline{t=12.5\ hours}}}$

Hence the tank gets filled in nearly $\bf{12.5\ hours.}$