Fitzpatrick 25 (e) (1 Viewer)

[DcM]

6. A vessel is being filled at a variable rate and the volume of liquid in the vessal at any time t is given by
V = A (1 - e^-kt)

a) show that dV/dt = k(A - V).........done already
b) if one quarter of the vessel is filled in 5 minutes, what fraction is filled in the next 5 minutes?

7. A tank contains 100 litres of brine whose concentration is 3 grams/litre. Three litres of brine whose concentration is 2 grams/litre flow into the tank each minute and at the same time 3 litres of mixture flow out each minute.
a) show that the quantity of salt, Q grams, in the tank at any time t is given by: Q = 200 + 100e^-0.03t

thanks!

 

[nit]

Is the answer for 6b:

(e^-5k)/4??

For 7:

The amount of salt entering the tank per minute is 3 x2 =6 grams per minute
Then the amount present at time t is Q(given). So the amount flowing out per minute is q/100 (ie the concentration) x 3 litres/minute (working the units out is really helpful in staying on the right track for these things)

So the instantaneous change in Q (ie the differential equation necessary) is dQ/dt = 6 - 3Q/100

Then just do LHS/RHS on the equation theyve provided for you, and it's all good

 

[nit]

Also, use cambridge maths if you can- theres no doubt that it's the best textbook for our maths course...(I reckon that the 3u yr 11 volume is the best textbook written by man )

 

[gman03]

Originally posted by DcM
b) if one quarter of the vessel is filled in 5 minutes, what fraction is filled in the next 5 minutes?

@ t = 5, V = A/4, so
1 - e^-5k = 1/4,
so e^-5k = 3/4

@ t = 10,
V = A(1 - e^-10k)
= A (1 - (e^-5k )²)
= A (1 - (3/4)²)
= A (1 - 9/16)
= 7/16 * A

So Change in V is 7A/16 - A/4 = 3A/16