p3 differential equations

Liquid is poured into a container at a constant rate of 30cm^3/s. At time t seconds liquid is leaking from the container at a rate of 2/15 V cm^3/s where V is the volume of liquid in the container at that time.

a) Show that -15 dV/dt = 2V - 450

Given that V=1000 when t=0

b) find the solution of the differential equation, in the form V=f(t)

c) Find the limiting value of V as t approaches infinity

Reply 1

yazan_l

a) increase in volume = 30
leaking = 2/15 V
dV/dt = 30 - 2/15 V, multiply by -15:
-15 dV/dt = 2V - 450

b) -15 dV/dt = 2V - 450
int -15/(2V-450) dV = int dt
-15/2 ln |2V-450| = t + C
-15/2 ln |2000-450| = 0 + C
C = -15/2 ln|1550|

-15/2 ln |2V-450| = t - 15/2 ln|1550|
15/2 ln| 1550 / (2V-450) | = t
1550 / (2V-450) = e^(2t/15)
1550e^(-2t/15) = (2V-450)
V = [1550e^(-2t/15) + 450 ]/2

c) for limiting: as t goes to infinity, 1550e^(-2t/15) goes to zero, so V goes to 450/2