Question Two

Thursday, 1 November 2018

6:03 PM

Untitled picture.png Machine generated alternative text:
i) The volume V of a tumour can be modelled by the differential equation 
dt 
where t is time, V is the volume of the tumour at time t and a and K 
are positive constants. If the initial value V (0) Vo is imposed, solving 
(*) as a separable equation gives the non constant solution 
(t 
l) 
a) Find all constant solutions to equation (*). 
b) Find the behaviour of V (t) as t —i 00. 
c) Give an interpretation of the constants a and K. 
d) Another model for tumour growth is given by the differential equation 
dt 
Suppose the same constants a and K are used in the two models. 
Without solving (**), explain which model predicts faster tumour 
growth for tumours when V is much smaller than K? 




































Untitled picture.png Machine generated alternative text:
ii) The specific gravity z of a solid heavier than water is given by 
a; 
where a; and y are its weight in air and water respectively. The weights 
a; and y are observed to be 21.3g and 10.2g and each observation is made 
with an uncertainty whose absolute value is at most 0. lg. 
Dz 
Dz 
a) Find — 
and 
Dy 
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Untitled picture.png Machine generated alternative text:
ii) The specific gravity z of a solid heavier than water is given by 
a; 
where a; and y are its weight in air and water respectively. The weights 
a; and y are observed to be 21.3g and 10.2g and each observation is made 
with an uncertainty whose absolute value is at most 0. lg. 
Dz 
Dz 
a) Find — 
and 
Dy 


































Untitled picture.png Machine generated alternative text:
b) Use the total differential approximation for z to estimate the maxi- 
mum uncertainty in the calculated value of z (to 3 decimal places). 
You may find the following Maple session useful. 
= x/ (x—y) : 
> zx := diff (z, x) 
:= diff (z, 
> zy 
> subs(x=21.3, 
> subs(x=21.3, 
y=10. 
y=10. 
2, 
2, 
zx) ; 
—0.08278548821 
zy) ; 
0.1728755783 
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Untitled picture.png Machine generated alternative text:
b) Use the total differential approximation for z to estimate the maxi- 
mum uncertainty in the calculated value of z (to 3 decimal places). 
You may find the following Maple session useful. 
= x/ (x—y) : 
> zx := diff (z, x) 
:= diff (z, 
> zy 
> subs(x=21.3, 
> subs(x=21.3, 
y=10. 
y=10. 
2, 
2, 
zx) ; 
—0.08278548821 
zy) ; 
0.1728755783 




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Machine generated alternative text:
iii) The probability density function f of a continuous random variable X is 
given by 
ka;2 
0 
where k is a constant. 
a) Find the value of k. 
b) Evaluate E(X) and Var(X). 
for 0 < x < 3 
otherwise,

 

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