Question

6. A tank is in the form of a right circular cylinder with hemispherical ends. The overall length of the tank is

4 meters and the diameter of the hemisphere is 1 meter. If a pump discharges a fluid whose density is

1.2 kg m /Liter in this tank at a rate of225 liters per minute determine a.) Weight of liquid inside the tank if

it is half full. b) total time to fill the tank assuming it is initially empty.

Answer 1

Let $d=$ the diameter of the hemisphere, $l=$the overall length of the tank

Find the volume of the hemisphere

$$V_1=\dfrac{1}{2}\big(\dfrac{4}{3}\pi(\dfrac{d}{2})^3\big)=\dfrac{\pi d^3}{12}$$

Find the volume of the right circular cylinder

$$V_2=\pi\big(\dfrac{d}{2}\big)^2(l-2(\dfrac{d}{2}))=\dfrac{\pi d^2(l-d)}{8}$$

Find the mass of liquid inside the tank if it is half full

$$m=\rho(V_1+V_2)=\rho\dfrac{\pi d^2(2d+3l-3d)}{24}$$

$$=\rho\dfrac{\pi d^2(3l-d)}{24}$$

$$m=1.2\ \dfrac{kgm}{10^{-3}\ m^3}\cdot\dfrac{\pi (1\ m)^2(3(4\ m)-1\ m)}{24}$$

$$=550\pi\ kgm\approx1728\ kg$$

Find the mass of liquid inside the tank if it is half full

$$W=mg$$

$$W=550\pi\ kgm(9.81\ m/s^2)\approx16950\ N$$

$$V=2(\dfrac{1}{2})\big(\dfrac{4}{3}\pi(\dfrac{d}{2})^3\big)=\dfrac{\pi d^3}{6}$$

b) Find the volume of the tank

$$V=2V_1+V_2$$

$$=2(\dfrac{\pi d^3}{12})+\dfrac{\pi d^2(l-d)}{8}$$

$$=\dfrac{\pi d^2(4d+3l-3d)}{24}$$

$$=\dfrac{\pi d^2(3l+d)}{24}$$

$$V=\dfrac{\pi (1\ m)^2(3(4\ m)+1\ m)}{24}=\dfrac{13\pi}{24}\ m^3$$

Find the total time to fill the tank assuming it is initially empty

$$t=\dfrac{\dfrac{13\pi}{24}\ m^3}{0.225 \ m^3/min}\approx7.6\ min$$