Question

A pool has a volume of 1000 ?3. At time ? = 0 the pool is empty. We then fill in water
the pool at a constant speed ? = 2.4?3 per minute. At the same time, water is leaking out
the pool at a rate that is at all times proportional to the volume of water.

The proportionality constant is ? = 3 ∙ 10−3??nute-1.

Let ? (?) be the volume of water in the pool t minutes after we started filling water in the pool.

a) Set up a differential equation that ? (?) must satisfy.
b) Solve the equation by the method with integrating factor.
c) Find when the pool is half full.

Answer 1

Assame that time ti amount of water in tanke = V Rate at which water enters Rin = a.4m m3/min Rate at which water leaves - Rout av * Bontserky, » Rout av, given 3 a = 3x10 3, * Rout 2 3x10 v m3/min 2 Rale in Rate out elv dt dv at 3 (8.4 3x10 v)

dv 2 3x10 V- 2.4 du + at du [ di + at spat .3 S 3x10 at 3x10 3x10 t PCU) v= QCV)' SP at Integrating factor.l.Fee Hege, entegrating factor = Solution Spdt sel spalt 3x10t QCA) di . v.e -3 3x10²4 Se e ve 2.4 dt 3x10²t 3x10² С + ve 2.4 Il -3 3x10

3x10 3x10 t v.e 800 e + C vCt) = 800+ cë 3x1õe when t=0, V=0 0 = 800+C > C=- 800 Solution' VG+) = 800 C1- ē3xlot) = 800 C1- 70.0035) (7)^ E CO) when poolis half full, vc+) = 500m3 R 500 - 800 (i- ē 0.003t) 5c 80.- 20.003+) 3/8 20.0036 1-58 nd Inloo =1-0.0036 -0.003t in (3/3) In(3/3) 11 to-) 0.003 to fill tank upto half . t- 326.94 minutes