Transcript Unit 1 Outcome 4 Recurrence Relations

Applications
Example 1
A hospital patient is put on medication which is taken once
per day. The dose is 35mg and each day the patient’s
metabolism burns off 70% of the drug in her system. It is
known that if the level of the drug in the patients system
reaches 54mg then the consequences could be fatal. Is it
safe for the patient to take the medication indefinitely?
***********************
We need to create a recurrence relation.
First dose = u0 = 35
Burning off 70% leaves behind 30% or 0.3.
After this another 35mg is taken so we have …..
un+1 = 0.3un + 35
This sequence has a limit since 0<0.3<1.
If we call the limit L then at this limit we have
un+1 = un = L
The equation
un+1 = 0.3un + 35 now becomes
L = 0.3L + 35
0.7L = 35
L = 35  0.7 = 350  7 = 50
Conclusion: the level of drug in the patients system will
never exceed 50mg under these conditions. Since this is
below the danger level it would be safe to continue
indefinitely.
Example 2 (trickier!)
The brake fluid reservoir in a car is leaky. Each day it
loses 3.14% of its contents. To compensate for this daily
loss the driver “tops up” once per week with 50ml of
fluid. For safety reasons the level of fluid in the reservoir
should always be between 200ml & 260ml. Initially it has
255ml.
(a) Find a recurrence relation to describe the above.
(b) Determine the fluid levels after 1 week and 4 weeks.
(c.) Is the process effective in the long run?
(a) Problem 3.14% daily loss = ? Weekly loss.
Losing 3.14% daily leaves behind 96.86% or 0.9686.
Amount remaining after 1 week = (0.9686)7 X A0
= 0.799854 X A0
= 0.80 X A0
or 80% of A0
This means that the car is losing 20% of its brake
fluid weekly.
So if An is the fluid level after n weeks then we have
An+1 = 0.8 An + 50
(b) Using
An+1 = 0.8 An + 50
with A0 = 255 we get
A0 = 255ml
A1 = 254ml
1st week (**)
A2 = 253.2ml
A3 = 252.6ml
A4 = 252.0ml
4th week
(**) even before adding the 50ml the level is above 200ml
(c) considering An+1 = 0.8 An + 50
Since 0<0.8<1 then a limit must exist and at this
An+1 = 0.8 An + 50
An+1=An= L so
ie
L = 0.8L + 50
or 0.2L = 50
or
L = 50  0.2 = 500  2 = 250
In the long run the weekly level will be 250ml and won’t fall
below 200ml so the driver should be OK with this routine.