2. You want to recreate Masterchef Australia’s recipe for mango daiquiris, court

Question

2. You want to recreate Masterchef Australia’s recipe for mango daiquiris, courtesy of Matt Preston, for a large party. Suppose a 500 litre tank initially contains 100 litres of chilled mineral water. A chilled mixture of white rum, lime juice, mango juice and water at 15% rum concentration is poured into the tank at 20 litres per minute. The mixture is stirred and simultaneously the mixture is withdrawn at 15 litres per minute.

(a) If x is the number of litres of rum in the tank at any time t minutes, show that dx=3 3x .

dt 20+t

(b) Find the amount of rum in the tank at any time.
(c) How long will it take for the mixture in the tank to reach a concentration of 10% rum?

(d) How long will it take for the tank to overflow?

Answer c and d please

Explanation / Answer

(a)x is the number of litres of rum in the tank at any time t minutes

rate of rum input =20*(15/100) =3 liters per minute

rate of rum output =15*x/(100+(20-15)t) = 15x/(100+5t) = 3x/(20+t)  liters per minute

net rate = rate of rum input -rate of rum output

(dx/dt)=3-(3x/(20+t))

initially there is no rum in tank .so x(0)=0

(b)

(dx/dt)+(3x/(20+t))=3

dx+(3x/(20+t))dt=3dt

integrating factor =e(3/(20+t))dt

integrating factor =e3ln(20+t)

integrating factor =(20+t)3

multiply on both sides by

dx(20+t)3+(3x/(20+t))(20+t)3dt=3(20+t)3dt

dx(20+t)3+ x3(20+t)2dt=3(20+t)3dt

=>(x(20+t)3)'=3(20+t)3dt

=>(x(20+t)3)=3(20+t)3dt

=>(x(20+t)3)=(3/4)(20+t)4+C

=>x=(3/4)(20+t) +(C/(20+t)3)

=>x=15+ (3/4)t +(C/(20+t)3)

x(0)=0

15+ (3/4)*0 +(C/(20+0)3) =0

=>15+(C/203) =0

=>C=-15*203

=>x=15+ (3/4)t +(-15*203/(20+t)3)

=>x=15+ (3/4)t -15(20/(20+t))3

amount of rum in the tank at any time. ,x=15+ (3/4)t -15(20/(20+t))3 or x=(3/4)(20+t) -15*(20/(20+t))3

(c)

concentration of 10% rum

=>(x/(100+5t)) =10/100

=>(((3/4)(20+t) -15*(20/(20+t)3))/(100+5t)) =10/100

=>(((3/20) -3*(203/(20+t)4)) =10/100

=>3*(203/(20+t)4)) =(3/20)-(10/100)

=>3*(203/(20+t)4)) =(1/20)

=>(20+t)4=3*204

=>(20+t)=20*31/4

=>t=20(31/4 -1)

=>t=6.32148

it takes 6.32 minutes for the mixture in the tank to reach a concentration of 10% rum

(d)

100+(20-15)t=500

=>100+5t=500

=>5t=400

=>t=80

it takes 80 minutes for tank to overflow

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