Page 40 - Pra U STPM 2022 Penggal 2 - Mathematics

Mathematics Semester 2 STPM Chapter 4 Differential Equations
8. The rate of decomposition of a radioactive substance is proportional to the mass of the remaining substance
at that time. If two-thirds of the initial mass remains after 100 days, find the time taken for one-half of
the initial mass to remain. Find the percentage of the mass that remains after 120 days.

9. Heat is supplied to an electric kettle at a constant rate of 2 000 watts whereas heat is lost to the surroundings
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at a rate of 20 watts for every Celsius degree difference between the temperature of the kettle and the
surrounding temperature. One watt of heat results in an increase in the temperature of the kettle at a rate
of 1 °C per minute. If the surrounding temperature is 15°C and θ°C is the temperature of the kettle after
50
t minutes, show that dθ = 40 – 2 (θ – 15).
dt 5
How long does it take for the temperature of the kettle to increase from 15°C to 100°C?

10. In a chemical reaction, two substances A and B react together to form another substance C. At time t, the
amount of A and B are a – x and b – x respectively, where a and b are constants and x is a function of
t. The value of x is zero when t = 0. At any time, the rate of decrease of A is proportional to the product
of the amount of A and B at that time. Find a differential equation involving x and t and solve x as a
function of t in each case of (i) a = b, (ii) a  b.

2
11. A water tank is in the shape of a cylinder with a vertical axis. The base area is A m . Initially, the tank
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is empty. Starting at time t = 0, water is poured into the tank at a constant rate of w m s , and water
–1
3
3
flows out from a small hole at the base at a rate of kx m s , where k is a constant and x m is the depth
of water in the tank. Form a differential equation and solve it to obtain a function of x in terms of t.
Show that no matter how long this process is repeated, the depth of the water will not exceed w m and
k
if the time taken to reach one-half of the depth of the cylinder is T s, then k = A ln 2.
T
4 12. An object moves along a straight line and passes a fixed point O with velocity u in the positive direction
of the x-axis. At time t the object is at a displacement x from O and the velocity of the object is v. The
rate of change of velocity has magnitude c v , where c is a constant, and is directed to towards the fixed
point O.
(a) Write down a differential equation for the motion of the object involving the velocity v and the time
t. Hence, find the velocity v as a function of the time t.
(b) Show that dv = v dv and hence, write down the differential equation for the motion of the object
dt dx
that involves the change of velocity v with respect to the displacement x and the velocity v.
Hence, find the velocity v as function of the displacement x.
2
—
3
Hence show that after a time t and the object has moved a distance x, 3cx = u – (u – 2ct) and deduce
2
3
2
that 2ct , u .
13. A research has been set up on an island to study a particular species of turtle. Initially there are 25 turtles
1
on the island. After t years the number of turtles x satisfies the differential equation dx = 20k x(k – x)
dt
where k is a constant.
(a) Show that k = 100 if it is known that the rate of growth is 0.45 turtle per year when x = 10.
(b) What is the maximum rate of growth?

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