A 200gallon tnak currently holds 30 pounds of salt dissolved

A 200-gallon tnak currently holds 30 pounds of salt dissolved in 50 gallons of water. Bring containing 2 pounds of salt per gallons runs into the tank at the rate of 2 gallons per minute, and the mixture, kept uniform by stirring. flows out at the rate of 1 gallons per minute.
(a)Set up and solve a differential equation for the amount of salt S(t) in solution at time t.
(b)How many gallons of solution are in the tank when there are 40 pounds of salt in solution?
(c) How much salt is in the tank when it overflows?

(a) let S(t) is the amount of salt present in the tank at time t

At t=0 ,

S(0) = 30 pounds

dS(t)/dt = 4 pounds/minute = 4 /60 pounds / sec = 1/15 pounds/sec

dS(t) = 1/15 dt (differential equation )

integrating on both sides

15S(t) = t + constant

at t=0 S(t) = 30

consatnt = 15 *30 = 450

a) S(t) = (t+450)/15

b) Given S(t) = 40

40 = (t + 450) /15

t= 150 s

total solution inthe tank T(t)

dT(t)/dt = 2/60 gallons/sec

integrating on both sides

30T(t) = t + consatnt

T(0) = 50

constant = 30*50 = 1500

T(t) = (t+1500)/30

GIven to find T(150)

T(150) = (150+1500)/30 = 55 gallons

b) Solution =55 gallons

c)to find t when T(t) = 200 gallons as the tank overflows when the tank is full and capavity of the tank is 200 gallons

200 = (t+1500)/30

t= 4500 s

S(4500) = (4500+45)/15

= 303 pounds

c) 303 pounds of salt

a) S(t) = (t+450)/15

b) Solution =55 gallons when 40 pounds of salt is present

c) 303 pounds of salt when overflows

A 200-gallon tnak currently holds 30 pounds of salt dissolved in 50 gallons of water. Bring containing 2 pounds of salt per gallons runs into the tank at the ra

A 200gallon tnak currently holds 30 pounds of salt dissolved

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