AboutAbe Mantell Expertise Hello, I am a college professor of mathematics and regularly teach all levels from elementary mathematics through differential equations, and would be happy to assist anyone with such questions!
Experience Over 15 years teaching at the college level.
Question An underground storage tank is being filled with liquid as shown in the diagram.Initially the tank is empty.At time t hours after filling begins,the volume of liquid is v m^3 & the depth of liquid is h m.It is given that v=4/3(h^3).The liquid is poured in at a rate of 20m^3 per hour,but owing to leakage,liquid is lost at a rate proportional to h^2.When h=1,dh/dt=4.95.i)show that h satisfies the differential equation dh/dt=5/h^2-1/(20). Many thanks
Answer dV/dt=20-k*h^2, based on the info given in the problem.
However, V=(4/3)h^3, thus, dV/dt=4*h^2*dh/dt. Now substitute that
into the first equation to get: 4*h^2*dh/dt=20-k*h^2. Dividing by
4*h^2 yields: dh/dt=5/h^2+k/4...now impose the condition: when h=1,
dh/dt=4.95 ==> 4.95=5+k/4. Thus, k/4=-0.05=-1/20. Hence, dh/dt=5/h^2-1/20.
