MTH108: HOMEWORK 7 – MODELLING USING DIFFERENTIAL EQUATIONS
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MTH108: HOMEWORK 7 – MODELLING USING DIFFERENTIAL EQUATIONS
Please attempt sections A and B before the tutorials on Thursdays.
Section A – Warm-up questions
A1. Solve the following first order initial value problems, gven that x(0) = x0 > 0:
(i) = sin(2t);
(ii) = t2 /x ;
(iii) = t(x + 1);
(iv) = −x2 ;
A2. The ambient water pressure, measured in atmospheres, is 1 at sea-level, and increases continuously at a rate of 1 atmosphere for every 10 metres below the surface . Write down a differential equation for the ambient pressure Pa as a function of the depth x and solve it .
Section B – Main questions
B1. In a chemical reaction, the mass m of a reactant decreases at a rate proportional to the square of its remaining mass . Half of the mass is used up in 20 minutes, so how much will be left after an hour?
B2. At 4am, when a police inspector arrives at a murder scene, the dead body’s temperature is 29 degrees . At 5am, when the inspector is ready to leave, the body’s temperature has fallen to 25 degrees . Assuming the victim’s temperature was 37 degrees before the murder and that the ambient temperature has remained a constant 21 degrees, use Newton’s Law of Cooling to estimate the time of death .
B3. A 20 litre tank is filled with a salt solution containing 20 grammes of salt . At time t = 0, a salt solution containing 2 grammes of salt per litre is added into the tank at a rate of 2 litres per minute . Solution is drained from the tank at the same rate . Assuming that the tank is stirred continuously, find and solve a differential equation for S(t), the mass of salt (in grammes) in the tank at time t (in minutes) .
B4. (Continued from A2.) A diver is planning to visit a coral reef that is at a depth of 20 metres . The pressure P(t) in the diver’s air tank (where t is the time in minutes) decreases at a rate proportional to the ambient pressure:
dP
(t) = −0 .3Pa (t) .
dt
Suppose the diver’s tank initially has a pressure of 200 atmospheres . The diver descends to the reef at a rate of 2 metres per minute, then remains at the reef for 10 minutes, and finally ascends to the surface at a rate of 1 metre per minute . Find the pressure in the tank at the end of the trip .
Section C – Extra problems
C1. Suppose we begin with a pure sample of a radioactive isotope X . The atoms of X decay to atoms of Y, which are themselves radioactive, decaying to atoms of Z . Write down differential equations for how the number of atoms of X and Y change with time, and solve them .
C2. (Continued from B4.) The partial pressure of nitrogen gas N(t) in the diver’s blood obeys
dN
= 0 .005[Pa (t) − N(t)]
dt
and initially we have N(0) = 0 .7 . Decompression sickness (”the bends”) can occur if N is ever greater than 3Pa during a dive . Determine how N(t) changes through the trip and check if the dive plan is safe .
2023-04-22