TS, oops, I did not read the question carefully.
This type of question will not appear in the actual "O" level
add maths exam.
This type of question will only appear in the H2 Maths
exam as it involves the use
of differential equation.
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Question (a)
In an experiment, the growth rate of bacteria in a liquid at any
time is proportional to N, the number of bacteria after
t seconds. Given that the initial number of bacteria is
A, show that N = Aekt , where
k is a constant.
Solution (copied and paste from Mr
Wee's answer)
dN/dt = kN, where k is the constant of proportionality.
Then we go on to solve by variable separable, i.e.
integral 1/N dN = integral k dt
ln N = kt + c, where c is a constant of integration
N = Ae^(kt), where A = e^c.
Question (b)(i)
After t minutes, the radius of a circle
r decreases at a rate of r/10 cm per minute and
the initial radius is 8 cm.
Express r in terms of
t.
Solution (copied and paste from Mr Wee's
answer)
differential equation dr/dt = -r/10.
By variable separable, integral 1/r dr = -1/10
integral dt
ln r = -0.1t + c, where c is the constant of
integration.
r = e ^(-0.1t + c)
r = e^c x e^(-0.1t)
Given that initial radius = 8 cm, so t = 0,
8 = e^c x e^(-0.1(0))
e^c = 8
Hence, r = e^c x e^(-0.1t)
=
8e^(-0.1t)
r = 8
e^(-t/10)
Question (b) (ii)
Find the time when the radius is 4 cm
Solution
when t = 4, r = 8 e^(-t/10)
4
= 8 e^(-t/10)
0.5
= e(-0.1t)
ln on both sides
ln 0.5 = ln e(-0.1t)
ln
0.5 = -0.1t
ln
0.5 / (- 0.1) = t
t
= 6.931471806
t
= 6 min 56 sec
