Answer:
[tex]\frac{dr}{dt} = 0.0535 ft/hr[/tex]
Step-by-step explanation:
Rate at which gas is escaping, dV/dt = 14 ft³/hr
Volume, V = 400 ft³
Volume of a sphere, [tex]V = \frac{4}{3} \pi r^{3}[/tex]................(1)
Differentiate both sides with respect to t
[tex]\frac{dV}{dt} = \frac{4}{3} \pi * 3r^{2}\frac{dr}{dt}[/tex]
[tex]\frac{dV}{dt} =4\pi r^{2} \frac{dr}{dt}[/tex]........................................(2)
From equation (1)
[tex]400 = \frac{4}{3} \pi r^{3}\\\\ r = (\frac{1200}{4\pi } )^{1/3} \\\\r = 4.564 ft[/tex]
Substitute the value of r and dV/dt into equation (2)
[tex]14 =4\pi * 4.564^{2} \frac{dr}{dt}\\\\\frac{14}{261.79} = \frac{dr}{dt}\\\\\frac{dr}{dt} = 0.0535 ft/hr[/tex]
