Model comparison/critique

1. In freezing temperatures, ice forms on the surface of the water in a barrel. At time \(t\) hours after the start of freezing, the thickness of the ice formed is \(x \mathrm {~mm}\). You may assume that the thickness of the ice is uniform across the surface of the water.

At 4 pm there is no ice on the surface, and freezing begins.
At 6pm, after two hours of freezing, the ice is 1.5 mm thick.
In a simple model, the rate of increase of \(x\), in mm per hour, is assumed to be constant for a period of 20 hours. Using this simple model,

1. express \(t\) in terms of \(x\),
2. find the value of \(t\) when \(x = 3\) In a second model, the rate of increase of \(x\), in mm per hour, is given by the differential equation $$\frac { \mathrm { d } x } { \mathrm {~d} t } = \frac { \lambda } { ( 2 x + 1 ) } \text { where } \lambda \text { is a constant and } 0 \leqslant t \leqslant 20$$ Using this second model,
3. solve the differential equation and express \(t\) in terms of \(x\) and \(\lambda\),
4. find the exact value for \(\lambda\),
5. find at what time the ice is predicted to be 3 mm thick.

8 The new price of a particular make of car is \(\pounds 10000\). When its age is \(t\) years, the list price is \(\pounds V\). When \(t = 5 , V = 5000\). Aloke, Ben and Charlie all run outlets for used cars. Each of them has a different model for the depreciation.

1. Aloke claims that the rate of depreciation is constant. Write this claim as a differential equation.
   Solve the differential equation and hence find the value of a car that is 7 years old according to this model.
   Explain why this model breaks down for large \(t\).
2. Ben believes that the rate of depreciation is inversely proportional to the square root of the age of the car. Express this claim as a differential equation and hence find the value of a car that is 7 years old according to this model.
   Does this model ever break down?
3. Charlie believes that a better model is given by the differential equation $$\frac { \mathrm { d } V } { \mathrm {~d} t } = k V$$ Solve this differential equation and find the value of the car after 7 years according to this model.
   Does this model ever break down?
4. Further investigation reveals that the average value of this particular type of car when 8 years old is \(\pounds 3000\). Find the value of \(V\) when \(t = 8\) for the three models above. Which of the three models best predicts the value of \(V\) at this time?

13 A scientist is attempting to model the number of insects, \(N\), present in a colony at time \(t\) weeks. When \(t = 0\) there are 400 insects and when \(t = 1\) there are 440 insects.

1. A scientist assumes that the rate of increase of the number of insects is inversely proportional to the number of insects present at time \(t\).
   1. Write down a differential equation to model this situation.
   2. Solve this differential equation to find \(N\) in terms of \(t\).
   3. In a revised model it is assumed that \(\frac { \mathrm { d } N } { \mathrm {~d} t } = \frac { N ^ { 2 } } { 3988 \mathrm { e } ^ { 0.2 t } }\). Solve this differential equation to find \(N\) in terms of \(t\).
   4. Compare the long-term behaviour of the two models.

12 A cake is cooling so that, \(t\) minutes after it is removed from an oven, its temperature is \(\theta ^ { \circ } \mathrm { C }\). When the cake is removed from the oven, its temperature is \(160 ^ { \circ } \mathrm { C }\). After 10 minutes its temperature has fallen to \(125 ^ { \circ } \mathrm { C }\).

1. In a simple model, the rate of decrease of the temperature of the cake is assumed to be constant.
   1. Write down a differential equation for this model.
   2. Solve this differential equation to find \(\theta\) in terms of \(t\).
   3. State one limitation of this model.
2. In a revised model, the rate of decrease of the temperature of the cake is proportional to the difference between the temperature of the cake and the temperature of the room. The temperature of the room is a constant \(20 ^ { \circ } \mathrm { C }\).
   1. Write down a differential equation for this revised model.
   2. Solve this differential equation to find \(\theta\) in terms of \(t\).
3. The cake can be decorated when its temperature is \(25 ^ { \circ } \mathrm { C }\). Find the difference in time between when the two models would predict that the cake can be decorated, giving your answer correct to the nearest minute. \section*{END OF QUESTION PAPER}

An entomologist is studying the population of insects in a colony. Initially there are 300 insects in the colony and in a model, the entomologist assumes that the population, \(P\), at time \(t\) weeks satisfies the differential equation $$\frac{dP}{dt} = kP,$$ where \(k\) is a constant.

1. Find an expression for \(P\) in terms of \(k\) and \(t\). [5]

Given that after one week there are 360 insects in the colony,

1. find the value of \(k\) to 3 significant figures. [2]

Given also that after two and three weeks there are 440 and 600 insects respectively,

1. comment on suitability of the model. [2]

An alternative model assumes that $$\frac{dP}{dt} = P(0.4 - 0.25\cos 0.5t).$$

1. Using the initial data, \(P = 300\) when \(t = 0\), solve this differential equation. [4]
2. Compare the suitability of the two models. [3]

An entomologist is studying the population of insects in a colony. Initially there are 300 insects in the colony and in a model, the entomologist assumes that the population, \(P\), at time \(t\) weeks satisfies the differential equation $$\frac{dP}{dt} = kP,$$ where \(k\) is a constant.

1. Find an expression for \(P\) in terms of \(k\) and \(t\). [5]

Given that after one week there are 360 insects in the colony,

1. find the value of \(k\) to 3 significant figures. [2]

Given also that after two and three weeks there are 440 and 600 insects respectively,

1. comment on suitability of the modelling assumption. [2]

An alternative model assumes that $$\frac{dP}{dt} = P(0.4 - 0.25 \cos 0.5t).$$

1. Using the initial data, \(P = 300\) when \(t = 0\), solve this differential equation. [3]
2. Compare the suitability of the two models. [2]

Sam goes on a diet. He assumes that his mass, \(m\) kg after \(t\) days, decreases at a rate that is inversely proportional to the cube root of his mass.

1. Construct a differential equation involving \(m\), \(t\) and a positive constant \(k\) to model this situation. [3 marks]
2. Explain why Sam's assumption may not be appropriate. [1 mark]