Page 39 - Pra U STPM 2022 Penggal 2 - Mathematics

Mathematics Semester 2 STPM Chapter 4 Differential Equations

Exercise 4.5

1. The height, h of a type of tree increases proportionately to the difference between its height at that time,
t and its final height, H. Find the differential equation connecting the variables h and t.
If h = 1 H when t = 0 and h = 1 H at t = T, solve the differential equation.
10 5
If time t is measured in days and T = 3, find the number of days passed before the height is more
than 9 H.
10

2. The manufacturer of a branded shampoo launches an advertisement program which results in the number
of consumers, n at time t to increase at a rate proportional to the square root of n. Write down the
differential equation which describes this relationship between n and t.
If n = N when t = 0 and n = 9 N when t = T, solve the differential equation. Find t when n = 4N.
4
3. Under certain conditions, the rate of cooling of a liquid is proportional to the difference between its
temperature and its surrounding temperature. The liquid is placed in a room of temperature 30°C and the
temperature of the liquid at time t (minutes) is x. Form a differential equation to describe this phenomena.
If the liquid cools from 100°C to 70°C in 8 minutes, find the further time taken for the liquid to reach a
temperature of 31°C. Find also the temperature of the liquid after it has been in the room for 10 minutes.

4. In a community, the number of people, n infected by a certain disease at a certain time increases at a rate
of λn. Form a differential equation connecting n and t.
λt
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Initially, the number of patients is N. Show that at day t, n = Ne . Find the value of λ if n = 2N when
t = 10. Find also
(a) the value of n when t = 20,
(b) the value of t when n = 3N.
4
5. In a certain chemical process, substance X is continuously changed to form substance Y. The total mass
of X and Y at any time is the same and equals to M. The rate of increase of Y at time t is proportional
to the mass of X at that time. If the mass of Y at time t is m, form a differential equation that describes
this chemical reaction. If M = 100 grams and 60 grams of substance X remains after 2 minutes, find the
mass of Y formed in 6 minutes.

6. At time t minutes, the number of micro-organisms in a liquid of a certain colony is x. The rate of increase
of x resulting from natural growth is αx and the rate of decrease due to death is β. Write down the
differential equation that describes the change in the size of the colony.
If the number of organisms is n at the beginning,
0
(a) find the time taken for all the micro-organisms in the colony to extinct if n = 200, α = 2 and
0
β = 500,
(b) find the number of micro-organisms in the colony after half a minute if n = 200, α = 4 and
0
β = 500.
7. At time t, the volume of water in a container is v. Due to leakage, water flows out at a rate of kv where k
is a positive constant. The rate at which water is lost due to evaporation is c. Obtain a differential equation
that describes the decrease in the volume of water. Given that v = V when t = 0, find the volume of water
in the container after t minutes. Find also the time taken before the container is empty.

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04 STPM Math(T) T2.indd 147 28/01/2022 5:44 PM