1.07t Construct differential equations: in context

7 When a stone is dropped into still water, ripples move outwards forming a circle of rippled water. At time \(t\) seconds after the stone hits the water the radius of the circle of ripples is increasing at a rate that is inversely proportional to the radius When the radius is 200 cm the rate of increase of the radius is 5 cm per second. Write down the differential equation that represents this situation.

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?

9 A liquid is being heated in an oven maintained at a constant temperature of \(160 ^ { \circ } \mathrm { C }\). It may be assumed that the rate of increase of the temperature of the liquid at any particular time \(t\) minutes is proportional to \(160 - \theta\), where \(\theta ^ { \circ } \mathrm { C }\) is the temperature of the liquid at that time.

1. Write down a differential equation connecting \(\theta\) and \(t\). When the liquid was placed in the oven, its temperature was \(20 ^ { \circ } \mathrm { C }\) and 5 minutes later its temperature had risen to \(65 ^ { \circ } \mathrm { C }\).
2. Find the temperature of the liquid, correct to the nearest degree, after another 5 minutes. 4

9 A tank contains water which is heated by an electric water heater working under the action of a thermostat. The temperature of the water, \(\theta ^ { \circ } \mathrm { C }\), may be modelled as follows. When the water heater is first switched on, \(\theta = 40\). The heater causes the temperature to increase at a rate \(k _ { 1 } { } ^ { \circ } \mathrm { C }\) per second, where \(k _ { 1 }\) is a constant, until \(\theta = 60\). The heater then switches off.

1. Write down, in terms of \(k _ { 1 }\), how long it takes for the temperature to increase from \(40 ^ { \circ } \mathrm { C }\) to \(60 ^ { \circ } \mathrm { C }\). The temperature of the water then immediately starts to decrease at a variable rate \(k _ { 2 } ( \theta - 20 ) ^ { \circ } \mathrm { C }\) per second, where \(k _ { 2 }\) is a constant, until \(\theta = 40\).
2. Write down a differential equation to represent the situation as the temperature is decreasing.
3. Find the total length of time for the temperature to increase from \(40 ^ { \circ } \mathrm { C }\) to \(60 ^ { \circ } \mathrm { C }\) and then decrease to \(40 ^ { \circ } \mathrm { C }\). Give your answer in terms of \(k _ { 1 }\) and \(k _ { 2 }\). 4

10 A container in the shape of an inverted cone of radius 3 metres and vertical height 4.5 metres is initially filled with liquid fertiliser. This fertiliser is released through a hole in the bottom of the container at a rate of \(0.01 \mathrm {~m} ^ { 3 }\) per second. At time \(t\) seconds the fertiliser remaining in the container forms an inverted cone of height \(h\) metres.
[0pt] [The volume of a cone is \(V = \frac { 1 } { 3 } \pi r ^ { 2 } h\).]

1. Show that \(h ^ { 2 } \frac { \mathrm {~d} h } { \mathrm {~d} t } = - \frac { 9 } { 400 \pi }\).
2. Express \(h\) in terms of \(t\).
3. Find the time it takes to empty the container, giving your answer to the nearest minute.

8 In the year 2000 the population density, \(P\), of a village was 100 people per \(\mathrm { km } ^ { 2 }\), and was increasing at the rate of 1 person per \(\mathrm { km } ^ { 2 }\) per year. The rate of increase of the population density is thought to be inversely proportional to the size of the population density. The time in years after the year 2000 is denoted by \(t\).

1. Write down a differential equation to model this situation, and solve it to express \(P\) in terms of \(t\).
2. In 2008 the population density of the village was 108 people per \(\mathrm { km } ^ { 2 }\) and in 2013 it was 128 people per \(\mathrm { km } ^ { 2 }\). Determine how well the model fits these figures.

7 Scientists can estimate the time elapsed since an animal died by measuring its body temperature.

1. Assuming the temperature goes down at a constant rate of 1.5 degrees Fahrenheit per hour, estimate how long it will take for the temperature to drop
   (A) from \(98 ^ { \circ } \mathrm { F }\) to \(89 ^ { \circ } \mathrm { F }\),
   (B) from \(98 ^ { \circ } \mathrm { F }\) to \(80 ^ { \circ } \mathrm { F }\). In practice, rate of temperature loss is not likely to be constant. A better model is provided by Newton's law of cooling, which states that the temperature \(\theta\) in degrees Fahrenheit \(t\) hours after death is given by the differential equation $$\frac { \mathrm { d } \theta } { \mathrm {~d} t } = - k \left( \theta - \theta _ { 0 } \right)$$ where \(\theta _ { 0 } { } ^ { \circ } \mathrm { F }\) is the air temperature and \(k\) is a constant.
2. Show by integration that the solution of this equation is \(\theta = \theta _ { 0 } + A \mathrm { e } ^ { - k t }\), where \(A\) is a constant. The value of \(\theta _ { 0 }\) is 50 , and the initial value of \(\theta\) is 98 . The initial rate of temperature loss is \(1.5 ^ { \circ } \mathrm { F }\) per hour.
3. Find \(A\), and show that \(k = 0.03125\).
4. Use this model to calculate how long it will take for the temperature to drop
   (A) from \(98 ^ { \circ } \mathrm { F }\) to \(89 ^ { \circ } \mathrm { F }\),
   (B) from \(98 ^ { \circ } \mathrm { F }\) to \(80 ^ { \circ } \mathrm { F }\).
5. Comment on the results obtained in parts (i) and (iv).

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.

8 A cylindrical tank is initially full of water. There is a small hole at the base of the tank out of which the water leaks. The height of water in the tank is \(x \mathrm {~m}\) at time \(t\) seconds. The rate of change of the height of water may be modelled by the assumption that it is proportional to the square root of the height of water. When \(t = 100 , x = 0.64\) and, at this instant, the height is decreasing at a rate of \(0.0032 \mathrm {~ms} ^ { - 1 }\).

1. Show that \(\frac { \mathrm { d } x } { \mathrm {~d} t } = - 0.004 \sqrt { x }\).
2. Find an expression for \(x\) in terms of \(t\).
3. Hence determine at what time, according to this model, the tank will be empty.

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}

6 Helga invests \(\pounds 4000\) in a savings account.
After \(t\) days, her investment is worth \(\pounds y\).
The rate of increase of \(y\) is \(k y\), where \(k\) is a constant.

1. Write down a differential equation in terms of \(t , y\) and \(k\).
2. Solve your differential equation to find the value of Helga's investment after \(t\) days. Give your answer in terms of \(k\) and \(t\). It is given that \(k = \frac { 1 } { 365 } \ln \left( 1 + \frac { r } { 100 } \right)\) where \(r \%\) is the rate of interest per annum. During the first year the rate of interest is \(6 \%\) per annum.
3. Find the value of Helga's investment after 90 days. After one year (365 days), the rate of interest drops to 5\% per annum.
4. Find the total time that it will take for Helga's investment to double in value.

11. In a science experiment, a radio active particle, \(N\), decays over time, \(t\), measured in minutes. The rate of decay of a particle is proportional to the number of particles remaining. Write down a suitable equation for the rate of change of the number of particles, \(N\) in terms of \(t\).

7. \begin{figure}[h] \end{figure} Figure 2 Figure 2 shows a cylindrical tank of height 1.5 m .
Initially the tank is full of water.
The water starts to leak from a small hole, at a point \(L\), in the side of the tank.
While the tank is leaking, the depth, \(H\) metres, of the water in the tank is modelled by the differential equation $$\frac { \mathrm { d } H } { \mathrm {~d} t } = - 0.12 \mathrm { e } ^ { - 0.2 t }$$ where \(t\) hours is the time after the leak starts.
Using the model,

1. show that $$H = A \mathrm { e } ^ { - 0.2 t } + B$$ where \(A\) and \(B\) are constants to be found,
2. find the time taken for the depth of the water to decrease to 1.2 m . Give your answer in hours and minutes, to the nearest minute. In the long term, the water level in the tank falls to the same height as the hole.
3. Find, according to the model, the height of the hole from the bottom of the tank.

1. A balloon is being inflated.

In a simple model,

• the balloon is modelled as a sphere
• the rate of increase of the radius of the balloon is inversely proportional to the square root of the radius of the balloon

At time \(t\) seconds, the radius of the balloon is \(r \mathrm {~cm}\).

1. Write down a differential equation to model this situation. At the instant when \(t = 10\)
   • the radius is 16 cm
   • the radius is increasing at a rate of \(0.9 \mathrm {~cm} \mathrm {~s} ^ { - 1 }\)
   • Solve the differential equation to show that
   $$r ^ { \frac { 3 } { 2 } } = 5.4 t + 10$$
2. Hence find the radius of the balloon when \(t = 20\) Give your answer to the nearest millimetre.
3. Suggest a limitation of the model.

1. A new smartphone was released by a company.

The company monitored the total number of phones sold, \(n\), at time \(t\) days after the phone was released. The company observed that, during this time,
the rate of increase of \(n\) was proportional to \(n\) Use this information to write down a suitable equation for \(n\) in terms of \(t\).
(You do not need to evaluate any unknown constants in your equation.)

1. A spherical mint of radius 5 mm is placed in the mouth and sucked. Four minutes later, the radius of the mint is 3 mm .

In a simple model, the rate of decrease of the radius of the mint is inversely proportional to the square of the radius. Using this model and all the information given,

1. find an equation linking the radius of the mint and the time.
   (You should define the variables that you use.)
2. Hence find the total time taken for the mint to completely dissolve. Give your answer in minutes and seconds to the nearest second.
3. Suggest a limitation of the model.

11. \begin{figure}[h] \end{figure} A tank in the shape of a cuboid is being filled with water.
The base of the tank measures 20 m by 10 m and the height of the tank is 5 m , as shown in Figure 1. At time \(t\) minutes after water started flowing into the tank the height of the water was \(h \mathrm {~m}\) and the volume of water in the tank was \(V \mathrm {~m} ^ { 3 }\) In a model of this situation

• the sides of the tank have negligible thickness
• the rate of change of \(V\) is inversely proportional to the square root of \(h\)
  1. Show that

$$\frac { \mathrm { d } h } { \mathrm {~d} t } = \frac { \lambda } { \sqrt { h } }$$ where \(\lambda\) is a constant. Given that

$$h ^ { \frac { 3 } { 2 } } = A t + B$$ where \(A\) and \(B\) are constants to be found.

Hence find the time taken, from when water started flowing into the tank, for the tank to be completely full.

14. \begin{figure}[h] \end{figure} Water flows at a constant rate into a large tank.
The tank is a cuboid, with all sides of negligible thickness.
The base of the tank measures 8 m by 3 m and the height of the tank is 5 m .
There is a tap at a point \(T\) at the bottom of the tank, as shown in Figure 5.
At time \(t\) minutes after the tap has been opened

• the depth of water in the tank is \(h\) metres
• water is flowing into the tank at a constant rate of \(0.48 \mathrm {~m} ^ { 3 }\) per minute
• water is modelled as leaving the tank through the tap at a rate of \(0.1 h \mathrm {~m} ^ { 3 }\) per minute
  1. Show that, according to the model,

$$1200 \frac { \mathrm {~d} h } { \mathrm {~d} t } = 24 - 5 h$$ Given that when the tap was opened, the depth of water in the tank was 2 m ,

show that, according to the model, $$h = A + B \mathrm { e } ^ { - k t }$$ where \(A , B\) and \(k\) are constants to be found. Given that the tap remains open,

determine, according to the model, whether the tank will ever become full, giving a reason for your answer.

1. A company decides to manufacture a soft drinks can with a capacity of 500 ml .

The company models the can in the shape of a right circular cylinder with radius \(r \mathrm {~cm}\) and height \(h \mathrm {~cm}\). In the model they assume that the can is made from a metal of negligible thickness.

1. Prove that the total surface area, \(S \mathrm {~cm} ^ { 2 }\), of the can is given by $$S = 2 \pi r ^ { 2 } + \frac { 1000 } { r }$$ Given that \(r\) can vary,
2. find the dimensions of a can that has minimum surface area.
3. With reference to the shape of the can, suggest a reason why the company may choose not to manufacture a can with minimum surface area.

1. A bacterial culture has area \(p \mathrm {~mm} ^ { 2 }\) at time \(t\) hours after the culture was placed onto a circular dish.

A scientist states that at time \(t\) hours, the rate of increase of the area of the culture can be modelled as being proportional to the area of the culture.

1. Show that the scientist's model for \(p\) leads to the equation $$p = a \mathrm { e } ^ { k t }$$ where \(a\) and \(k\) are constants. The scientist measures the values for \(p\) at regular intervals during the first 24 hours after the culture was placed onto the dish. She plots a graph of \(\ln p\) against \(t\) and finds that the points on the graph lie close to a straight line with gradient 0.14 and vertical intercept 3.95
2. Estimate, to 2 significant figures, the value of \(a\) and the value of \(k\).
3. Hence show that the model for \(p\) can be rewritten as $$p = a b ^ { t }$$ stating, to 3 significant figures, the value of the constant \(b\). With reference to this model,
4. 1. interpret the value of the constant \(a\),
   2. interpret the value of the constant \(b\).
5. State a long term limitation of the model for \(p\).

11 A balloon is being inflated. The balloon is modelled as a sphere with radius \(x \mathrm {~cm}\) at time \(t \mathrm {~s}\). The volume \(V \mathrm {~cm} ^ { 3 }\) is given by \(\mathrm { V } = \frac { 4 } { 3 } \pi \mathrm { x } ^ { 3 }\). The rate of increase of volume is inversely proportional to the radius of the balloon. Initially, when \(t = 0\), the radius of the balloon is 5 cm and the volume of the balloon is increasing at a rate of \(21 \mathrm {~cm} ^ { 3 } \mathrm {~s} ^ { - 1 }\).

1. Show that \(x\) satisfies the differential equation \(\frac { \mathrm { dx } } { \mathrm { dt } } = \frac { 105 } { 4 \pi \mathrm { x } ^ { 3 } }\).
2. Find the radius of the balloon after two minutes.
3. Explain why the model may not be suitable for very large values of \(t\).

6 A hot drink is cooling. The temperature of the drink at time \(t\) minutes is \(T ^ { \circ } \mathrm { C }\).
The rate of decrease in temperature of the drink is proportional to \(( T - 20 )\).

1. Write down a differential equation to describe the temperature of the drink as a function of time.
2. When \(t = 0\), the temperature of the drink is \(90 ^ { \circ } \mathrm { C }\) and the temperature is decreasing at a rate of \(4.9 ^ { \circ } \mathrm { C }\) per minute. Determine how long it takes for the drink to cool from \(90 ^ { \circ } \mathrm { C }\) to \(40 ^ { \circ } \mathrm { C }\).

7 A giant snowball is melting. The snowball can be modelled as a sphere whose surface area is decreasing at a constant rate with respect to time. The surface area of the sphere is \(A \mathrm {~cm} ^ { 2 }\) at time \(t\) days after it begins to melt.

1. Write down a differential equation in terms of the variables \(A\) and \(t\) and a constant \(k\), where \(k > 0\), to model the melting snowball.
2. 1. Initially, the radius of the snowball is 60 cm , and 9 days later, the radius has halved. Show that \(A = 1200 \pi ( 12 - t )\).
      (You may assume that the surface area of a sphere is given by \(A = 4 \pi r ^ { 2 }\), where \(r\) is the radius.)
   2. Use this model to find the number of days that it takes the snowball to melt completely.

8

1. A water tank has a height of 2 metres. The depth of the water in the tank is \(h\) metres at time \(t\) minutes after water begins to enter the tank. The rate at which the depth of the water in the tank increases is proportional to the difference between the height of the tank and the depth of the water. Write down a differential equation in the variables \(h\) and \(t\) and a positive constant \(k\).
   (You are not required to solve your differential equation.)
2. 1. Another water tank is filling in such a way that \(t\) minutes after the water is turned on, the depth of the water, \(x\) metres, increases according to the differential equation $$\frac { \mathrm { d } x } { \mathrm {~d} t } = \frac { 1 } { 15 x \sqrt { 2 x - 1 } }$$ The depth of the water is 1 metre when the water is first turned on.
      Solve this differential equation to find \(t\) as a function of \(x\).
   2. Calculate the time taken for the depth of the water in the tank to reach 2 metres, giving your answer to the nearest 0.1 of a minute.
      (l mark)

7 The height of the tide in a certain harbour is \(h\) metres at time \(t\) hours. Successive high tides occur every 12 hours. The rate of change of the height of the tide can be modelled by a function of the form \(a \cos ( k t )\), where \(a\) and \(k\) are constants. The largest value of this rate of change is 1.3 metres per hour. Write down a differential equation in the variables \(h\) and \(t\). State the values of the constants \(a\) and \(k\).