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

Ink Drawings
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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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