1.07t Construct differential equations: in context

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1. A pond is initially empty and is then filled gradually with water. After \(t\) minutes, the depth of the water, \(x\) metres, satisfies the differential equation $$\frac { \mathrm { d } x } { \mathrm {~d} t } = \frac { \sqrt { 4 + 5 x } } { 5 ( 1 + t ) ^ { 2 } }$$ Solve this differential equation to find \(x\) in terms of \(t\).
2. Another pond is gradually filling with water. After \(t\) minutes, the surface of the water forms a circle of radius \(r\) metres. The rate of change of the radius is inversely proportional to the area of the surface of the water.
   1. Write down a differential equation, in the variables \(r\) and \(t\) and a constant of proportionality, which represents how the radius of the surface of the water is changing with time.
      (You are not required to solve your differential equation.)
   2. When the radius of the pond is 1 metre, the radius is increasing at a rate of 4.5 metres per second. Find the radius of the pond when the radius is increasing at a rate of 0.5 metres per second.
      [0pt] [2 marks]
      \includegraphics[max width=\textwidth, alt={}]{fdd3905e-11f7-4b20-adfe-4c686018a221-18_1277_1709_1430_153}
      \includegraphics[max width=\textwidth, alt={}]{fdd3905e-11f7-4b20-adfe-4c686018a221-20_2288_1707_221_153}

5. A bath is filled with hot water which is allowed to cool. The temperature of the water is \(\theta ^ { \circ } \mathrm { C }\) after cooling for \(t\) minutes and the temperature of the room is assumed to remain constant at \(20 ^ { \circ } \mathrm { C }\). Given that the rate at which the temperature of the water decreases is proportional to the difference in temperature between the water and the room,

1. write down a differential equation connecting \(\theta\) and \(t\). Given also that the temperature of the water is initially \(37 ^ { \circ } \mathrm { C }\) and that it is \(36 ^ { \circ } \mathrm { C }\) after cooling for four minutes,
2. find, to 3 significant figures, the temperature of the water after ten minutes. Advice suggests that the temperature of the water should be allowed to cool to \(33 ^ { \circ } \mathrm { C }\) before a child gets in.
3. Find, to the nearest second, how long a child should wait before getting into the bath.
   5. continued

6 A car accelerates from rest along a straight horizontal road. The car's engine produces a constant horizontal force of magnitude 4000 N .
At time \(t\) seconds, the speed of the car is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), and a resistance force of magnitude \(40 v\) newtons acts upon the car. The mass of the car is 1600 kg .

1. Show that \(\frac { \mathrm { d } v } { \mathrm {~d} t } = \frac { 100 - v } { 40 }\).
2. Find the velocity of the car at time \(t\).

7 A parachutist, of mass 72 kg , is falling vertically. He opens his parachute at time \(t = 0\) when his speed is \(30 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). He then experiences an air resistance force of magnitude \(240 v\) newtons, where \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) is his speed at time \(t\) seconds.

1. When \(t > 0\), show that \(- \frac { 3 } { 10 } \frac { \mathrm {~d} v } { \mathrm {~d} t } = v - 2.94\).
2. Find \(v\) in terms of \(t\).
3. Sketch a graph to show how, for \(t \geqslant 0\), the parachutist's speed varies with time.
   [0pt] [2 marks]

1 A curve passes through the point ( 1,3 ) and satisfies the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } = 1 + x ^ { 3 }$$ Starting at the point ( 1,3 ), use a step-by-step method with a step length of 0.1 to estimate the \(y\)-coordinate of the point on the curve for which \(x = 1.3\). Give your answer to three decimal places.
(No credit will be given for methods involving integration.)

9. The diagram below shows a parcel, of mass \(m \mathrm {~kg}\), sliding down a rough slope inclined at an angle \(\alpha\) to the horizontal, where \(\sin \alpha = \frac { 7 } { 25 }\). \includegraphics[max width=\textwidth, alt={}, center]{8f47b2ff-f954-42ec-8ecc-fc64313a7b89-24_394_906_497_584} The coefficient of friction between the parcel and the slope is \(\frac { 1 } { 12 }\). In addition to friction, the parcel experiences a variable resistive force of \(m v \mathrm {~N}\), where \(v \mathrm {~ms} ^ { - 1 }\) is the velocity of the parcel at time \(t\) seconds.

1. Show that the motion of the parcel satisfies the differential equation $$5 \frac { \mathrm {~d} v } { \mathrm {~d} t } = g - 5 v$$

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4. \begin{figure}[h] \end{figure} A manufacturer produces cartons for fruit juice. Each carton is in the shape of a closed cuboid with base dimensions \(2 x \mathrm {~cm}\) by \(x \mathrm {~cm}\) and height \(h \mathrm {~cm}\), as shown in Fig. 4. Given that the capacity of a carton has to be \(1030 \mathrm {~cm} ^ { 3 }\),

1. express \(h\) in terms of \(x\),
2. show that the surface area, \(A \mathrm {~cm} ^ { 2 }\), of a carton is given by $$A = 4 x ^ { 2 } + \frac { 3090 } { x } .$$

8 A disease is spreading through a colony of rabbits. There are 5000 rabbits in the colony. At time \(t\) hours, \(x\) is the number of rabbits infected. The rate of increase of the number of rabbits infected is proportional to the product of the number of rabbits infected and the number not yet infected.

1. 1. Formulate a differential equation for \(\frac { \mathrm { d } x } { \mathrm {~d} t }\) in terms of the variables \(x\) and \(t\) and a constant of proportionality \(k\).
   2. Initially, 1000 rabbits are infected and the disease is spreading at a rate of 200 rabbits per hour. Find the value of the constant \(k\).
      (You are not required to solve your differential equation.)
2. The solution of the differential equation in this model is $$t = 4 \ln \left( \frac { 4 x } { 5000 - x } \right)$$
   1. Find the time after which 2500 rabbits will be infected, giving your answer in hours to one decimal place.
   2. Find, according to this model, the number of rabbits infected after 30 hours.

8

1. The number of fish in a lake is decreasing. After \(t\) years, there are \(x\) fish in the lake. The rate of decrease of the number of fish is proportional to the number of fish currently in the lake.
   1. Formulate a differential equation, in the variables \(x\) and \(t\) and a constant of proportionality \(k\), where \(k > 0\), to model the rate at which the number of fish in the lake is decreasing.
   2. At a certain time, there were 20000 fish in the lake and the rate of decrease was 500 fish per year. Find the value of \(k\).
2. The equation $$P = 2000 - A \mathrm { e } ^ { - 0.05 t }$$ is proposed as a model for the number of fish, \(P\), in another lake, where \(t\) is the time in years and \(A\) is a positive constant. On 1 January 2008, a biologist estimated that there were 700 fish in this lake.
   1. Taking 1 January 2008 as \(t = 0\), find the value of \(A\).
   2. Hence find the year during which, according to this model, the number of fish in this lake will first exceed 1900.

5. \begin{figure}[h] \end{figure} Figure 3 shows a container in the shape of a hollow, inverted, right circular cone.
The height of the container is 30 cm and the radius is 12 cm , as shown in Figure 3.
The container is initially empty when water starts flowing into it.
When the height of water is \(h \mathrm {~cm}\), the surface of the water has radius \(r \mathrm {~cm}\) and the volume of water is \(V \mathrm {~cm} ^ { 3 }\)

1. Show that $$V = \frac { 4 \pi h ^ { 3 } } { 75 }$$ [The volume \(V\) of a right circular cone with vertical height \(h\) and base radius \(r\) is given by the formula \(V = \frac { 1 } { 3 } \pi r ^ { 2 } h\) ] Given that water flows into the container at a constant rate of \(2 \pi \mathrm {~cm} ^ { 3 } \mathrm {~s} ^ { - 1 }\)
2. find, in \(\mathrm { cm } \mathrm { s } ^ { - 1 }\), the rate at which \(h\) is changing, exactly 1.5 minutes after water starts flowing into the container.

A rectangle is inscribed in a semicircle with centre \(O\) and radius 4. The point \(P ( x , y )\) is the vertex of the rectangle in the first quadrant as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{72bb1603-edbd-4e2e-bf2b-f33bb667e61b-6_553_929_1430_593}
a) Express the area \(A\) of the rectangle as a function of \(x\).
b) Show that the maximum value of \(A\) occurs when \(y = x\).

\includegraphics{figure_10} A tank containing water is in the form of a cone with vertex \(C\). The axis is vertical and the semi-vertical angle is \(60°\), as shown in the diagram. At time \(t = 0\), the tank is full and the depth of water is \(H\). At this instant, a tap at \(C\) is opened and water begins to flow out. The volume of water in the tank decreases at a rate proportional to \(\sqrt{h}\), where \(h\) is the depth of water at time \(t\). The tank becomes empty when \(t = 60\).

1. Show that \(h\) and \(t\) satisfy a differential equation of the form $$\frac{dh}{dt} = -Ah^{-\frac{1}{2}},$$ where \(A\) is a positive constant. [4]
2. Solve the differential equation given in part (i) and obtain an expression for \(t\) in terms of \(h\) and \(H\). [6]
3. Find the time at which the depth reaches \(\frac{1}{2}H\). [1]

[The volume \(V\) of a cone of vertical height \(h\) and base radius \(r\) is given by \(V = \frac{1}{3}\pi r^2 h\).]

A spherical ball of ice of radius 12 cm is placed in a bucket of water. In a model of the situation, • the ball remains spherical as it melts • \(t\) minutes after the ball of ice is placed in the bucket, its radius is \(r\) cm • the rate of decrease of the radius of the ball of ice is inversely proportional to the square of the radius • the radius of the ball of ice is 6 cm after 15 minutes Using the model and the information given,

1. find an equation linking \(r\) and \(t\), [5]
2. find the time taken for the ball of ice to melt completely, [2]
3. On Diagram 1 on page 27, sketch a graph of \(r\) against \(t\). [1]

\includegraphics{figure_4} A manufacturer produces cartons for fruit juice. Each carton is in the shape of a closed cuboid with base dimensions 2\(x\) cm by \(x\) cm and height \(h\) cm, as shown in Fig. 4. Given that the capacity of a carton has to be 1030 cm\(^3\),

1. express \(h\) in terms of \(x\), [2]
2. show that the surface area, \(A\) cm\(^2\), of a carton is given by $$A = 4x^2 + \frac{3090}{x}.$$ [3]

A forest is burning so that, \(t\) hours after the start of the fire, the area burnt is \(A\) hectares. It is given that, at any instant, the rate at which this area is increasing is proportional to \(A^2\).

1. Write down a differential equation which models this situation. [2]
2. After 1 hour, 1000 hectares have been burnt; after 2 hours, 2000 hectares have been burnt. Find after how many hours 3000 hectares have been burnt. [6]

The total value of the sales made by a new company in the first \(t\) years of its existence is denoted by \(£V\). A model is proposed in which the rate of increase of \(V\) is proportional to the square root of \(V\). The constant of proportionality is \(k\).

1. Express the model as a differential equation. Verify by differentiation that \(V = (\frac{1}{2}kt + c)^2\), where \(c\) is an arbitrary constant, satisfies this differential equation. [4]
2. The value of the company's sales in its first year is £10000, and the total value of the sales in the first two years is £40000. Find \(V\) in terms of \(t\). [4]

At time \(t = 0\), a tank of height 2 metres is completely filled with water. Water then leaks from a hole in the side of the tank such that the depth of water in the tank, \(y\) metres, after \(t\) hours satisfies the differential equation $$\frac{dy}{dt} = -ke^{-0.2t},$$ where \(k\) is a positive constant.

1. Find an expression for \(y\) in terms of \(k\) and \(t\). [4]

Given that two hours after being filled the depth of water in the tank is 1.6 metres,

1. find the value of \(k\) to 4 significant figures. [2]

Given also that the hole in the tank is \(h\) cm above the base of the tank,

1. show that \(h = 79\) to 2 significant figures. [3]

The rate of increase in the number of bacteria in a culture, \(N\), at time \(t\) hours is proportional to \(N\).

1. Write down a differential equation connecting \(N\) and \(t\). [1]

Given that initially there are \(N_0\) bacteria present in a culture,

1. Show that \(N = N_0 e^{kt}\), where \(k\) is a positive constant. [6]

Given also that the number of bacteria present doubles every six hours,

1. find the value of \(k\), [3]
2. Find how long it takes for the number of bacteria to increase by a factor of ten, giving your answer to the nearest minute. [2]

A gardener stores rainwater in a cylindrical container. The container has a height of 130 centimetres. The gardener empties the water from the container through a hose. The hose is attached 5 centimetres from the bottom of the container. At time \(t\) minutes after the hose is switched on, the depth of water, \(h\) centimetres, in the container decreases at a rate which is proportional to \(h - 5\) Initially the container of water is full, and the depth of water is decreasing at a rate of 1.5 centimetres per minute.

1. Show that $$\frac{dh}{dt} = -0.012(h - 5)$$ [3 marks]
2. Solve the differential equation $$\frac{dh}{dt} = -0.012(h - 5)$$ to find an expression for \(h\) in terms of \(t\) [5 marks]
3. Find the time taken for the container to be half empty. Give your answer to the nearest minute. [2 marks]

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]

A market trader notices that daily sales are dependent on two variables: number of hours, \(t\), after the stall opens total sales, \(x\), in pounds since the stall opened. The trader models the rate of sales as directly proportional to \(\frac{8 - t}{x}\) After two hours the rate of sales is £72 per hour and total sales are £336

1. Show that $$x \frac{dx}{dt} = 4032(8 - t)$$ [3 marks]
2. Hence, show that $$x^2 = 4032t(16 - t)$$ [3 marks]
3. The stall opens at 09.30.
   1. The trader closes the stall when the rate of sales falls below £24 per hour. Using the results in parts (a) and (b), calculate the earliest time that the trader closes the stall. [6 marks]
   2. Explain why the model used by the trader is not valid at 09.30. [2 marks]

A particle moves so that its acceleration, \(a\text{ ms}^{-2}\), at time \(t\) seconds may be modelled in terms of its velocity, \(v\text{ ms}^{-1}\), as $$a = -0.1v^2$$ The initial velocity of the particle is \(4\text{ ms}^{-1}\)

1. By first forming a suitable differential equation, show that $$v = \frac{20}{5 + 2t}$$ [6 marks]
2. Find the acceleration of the particle when \(t = 5.5\) [2 marks]

The rate of change of a variable \(y\) with respect to \(x\) is directly proportional to \(y\).

1. Write down a differential equation satisfied by \(y\). [1]
2. When \(x = 1\) and \(y = 0.5\), the rate of change of \(y\) with respect to \(x\) is 2. Find \(y\) when \(x = 3\). [6]