1.02z Models in context: use functions in modelling

9 The volume \(V \mathrm {~m} ^ { 3 }\) of a large circular mound of iron ore of radius \(r \mathrm {~m}\) is modelled by the equation \(V = \frac { 3 } { 2 } \left( r - \frac { 1 } { 2 } \right) ^ { 3 } - 1\) for \(r \geqslant 2\). Iron ore is added to the mound at a constant rate of \(1.5 \mathrm {~m} ^ { 3 }\) per second.
[0pt]

1. Find the rate at which the radius of the mound is increasing at the instant when the radius is 5.5 m . [3]
2. Find the volume of the mound at the instant when the radius is increasing at 0.1 m per second.

9 The base of a cuboid has sides of length \(x \mathrm {~cm}\) and \(3 x \mathrm {~cm}\). The volume of the cuboid is \(288 \mathrm {~cm} ^ { 3 }\).

1. Show that the total surface area of the cuboid, \(A \mathrm {~cm} ^ { 2 }\), is given by $$A = 6 x ^ { 2 } + \frac { 768 } { x }$$
2. Given that \(x\) can vary, find the stationary value of \(A\) and determine its nature.

6 A vacuum flask (for keeping drinks hot) is modelled as a closed cylinder in which the internal radius is \(r \mathrm {~cm}\) and the internal height is \(h \mathrm {~cm}\). The volume of the flask is \(1000 \mathrm {~cm} ^ { 3 }\). A flask is most efficient when the total internal surface area, \(A \mathrm {~cm} ^ { 2 }\), is a minimum.

1. Show that \(A = 2 \pi r ^ { 2 } + \frac { 2000 } { r }\).
2. Given that \(r\) can vary, find the value of \(r\), correct to 1 decimal place, for which \(A\) has a stationary value and verify that the flask is most efficient when \(r\) takes this value.

3 \includegraphics[max width=\textwidth, alt={}, center]{f759ce41-708e-4fe7-80b9-adc2be2972ac-04_489_465_258_840} The diagram shows a water container in the form of an inverted pyramid, which is such that when the height of the water level is \(h \mathrm {~cm}\) the surface of the water is a square of side \(\frac { 1 } { 2 } h \mathrm {~cm}\).

1. Express the volume of water in the container in terms of \(h\).
   [0pt] [The volume of a pyramid having a base area \(A\) and vertical height \(h\) is \(\frac { 1 } { 3 } A h\).]
   Water is steadily dripping into the container at a constant rate of \(20 \mathrm {~cm} ^ { 3 }\) per minute.
2. Find the rate, in cm per minute, at which the water level is rising when the height of the water level is 10 cm . \includegraphics[max width=\textwidth, alt={}, center]{f759ce41-708e-4fe7-80b9-adc2be2972ac-06_403_773_258_685} In the diagram, \(A B = A C = 8 \mathrm {~cm}\) and angle \(C A B = \frac { 2 } { 7 } \pi\) radians. The circular \(\operatorname { arc } B C\) has centre \(A\), the circular arc \(C D\) has centre \(B\) and \(A B D\) is a straight line.

9 \includegraphics[max width=\textwidth, alt={}, center]{dd2cb0ec-5df9-4d99-9e15-5ae1f1c07b96-4_387_903_799_623} The diagram shows an open container constructed out of \(200 \mathrm {~cm} ^ { 2 }\) of cardboard. The two vertical end pieces are isosceles triangles with sides \(5 x \mathrm {~cm} , 5 x \mathrm {~cm}\) and \(8 x \mathrm {~cm}\), and the two side pieces are rectangles of length \(y \mathrm {~cm}\) and width \(5 x \mathrm {~cm}\), as shown. The open top is a horizontal rectangle.

1. Show that \(y = \frac { 200 - 24 x ^ { 2 } } { 10 x }\).
2. Show that the volume, \(V \mathrm {~cm} ^ { 3 }\), of the container is given by \(V = 240 x - 28.8 x ^ { 3 }\). Given that \(x\) can vary,
3. find the value of \(x\) for which \(V\) has a stationary value,
4. determine whether it is a maximum or a minimum stationary value.

7 \includegraphics[max width=\textwidth, alt={}, center]{08729aab-586b-4210-94c9-77b1f6b1d873-3_385_360_1379_561} \includegraphics[max width=\textwidth, alt={}, center]{08729aab-586b-4210-94c9-77b1f6b1d873-3_364_369_1379_1219} A wire, 80 cm long, is cut into two pieces. One piece is bent to form a square of side \(x \mathrm {~cm}\) and the other piece is bent to form a circle of radius \(r \mathrm {~cm}\) (see diagram). The total area of the square and the circle is \(A \mathrm {~cm} ^ { 2 }\).

1. Show that \(A = \frac { ( \pi + 4 ) x ^ { 2 } - 160 x + 1600 } { \pi }\).
2. Given that \(x\) and \(r\) can vary, find the value of \(x\) for which \(A\) has a stationary value.

7 In a certain chemical process a substance is being formed, and \(t\) minutes after the start of the process there are \(m\) grams of the substance present. In the process the rate of increase of \(m\) is proportional to \(( 50 - m ) ^ { 2 }\). When \(t = 0 , m = 0\) and \(\frac { \mathrm { d } m } { \mathrm {~d} t } = 5\).

1. Show that \(m\) satisfies the differential equation $$\frac { \mathrm { d } m } { \mathrm {~d} t } = 0.002 ( 50 - m ) ^ { 2 }$$
2. Solve the differential equation, and show that the solution can be expressed in the form $$m = 50 - \frac { 500 } { t + 10 }$$
3. Calculate the mass of the substance when \(t = 10\), and find the time taken for the mass to increase from 0 to 45 grams.
4. State what happens to the mass of the substance as \(t\) becomes very large.

8 In a certain chemical reaction, a compound \(A\) is formed from a compound \(B\). The masses of \(A\) and \(B\) at time \(t\) after the start of the reaction are \(x\) and \(y\) respectively and the sum of the masses is equal to 50 throughout the reaction. At any time the rate of increase of the mass of \(A\) is proportional to the mass of \(B\) at that time.

1. Explain why \(\frac { \mathrm { d } x } { \mathrm {~d} t } = k ( 50 - x )\), where \(k\) is a constant.
   It is given that \(x = 0\) when \(t = 0\), and \(x = 25\) when \(t = 10\).
2. Solve the differential equation in part (i) and express \(x\) in terms of \(t\).

10 \includegraphics[max width=\textwidth, alt={}, center]{dd7b2aee-4318-48e8-97c0-541e47f2e83a-4_335_875_262_635} A tank containing water is in the form of a cone with vertex \(C\). The axis is vertical and the semivertical angle is \(60 ^ { \circ }\), 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 $$\frac { \mathrm { d } h } { \mathrm {~d} t } = - A h ^ { - \frac { 3 } { 2 } } ,$$ where \(A\) is a positive constant.
2. Solve the differential equation given in part (i) and obtain an expression for \(t\) in terms of \(h\) and \(H\).
3. Find the time at which the depth reaches \(\frac { 1 } { 2 } H\).
   [0pt] [The volume \(V\) of a cone of vertical height \(h\) and base radius \(r\) is given by \(V = \frac { 1 } { 3 } \pi r ^ { 2 } h\).]

7 In a certain country the government charges tax on each litre of petrol sold to motorists. The revenue per year is \(R\) million dollars when the rate of tax is \(x\) dollars per litre. The variation of \(R\) with \(x\) is modelled by the differential equation $$\frac { \mathrm { d } R } { \mathrm {~d} x } = R \left( \frac { 1 } { x } - 0.57 \right)$$ where \(R\) and \(x\) are taken to be continuous variables. When \(x = 0.5 , R = 16.8\).

1. Solve the differential equation and obtain an expression for \(R\) in terms of \(x\).
2. This model predicts that \(R\) cannot exceed a certain amount. Find this maximum value of \(R\).

1. A tree was planted.

Exactly 3 years after it was planted, the height of the tree was 2 m . Exactly 5 years after it was planted, the height of the tree was 2.4 m . Given that the height, \(H\) metres, of the tree, \(t\) years after it was planted, can be modelled by the equation $$H ^ { 3 } = p t ^ { 2 } + q$$ where \(p\) and \(q\) are constants,

1. find, to 3 significant figures where necessary, the value of \(p\) and the value of \(q\). Exactly \(T\) years after the tree was planted, its height was 5 m .
2. Find the value of \(T\) according to the model, giving your answer to one decimal place.

2. A tree was planted in the ground. Exactly 2 years after it was planted, the height of the tree was 1.85 m . Exactly 7 years after it was planted, the height of the tree was 3.45 m . Given that the height, \(H\) metres, of the tree, \(t\) years after it was planted in the ground, can be modelled by the equation $$H = a t + b$$ where \(a\) and \(b\) are constants,

1. find the value of \(a\) and the value of \(b\).
2. State, according to the model, the height of the tree when it was planted.
   

7. \begin{figure}[h] \end{figure} Figure 1 shows a rectangular sheet of metal of negligible thickness, which measures 25 cm by 15 cm . Squares of side \(x \mathrm {~cm}\) are cut from each corner of the sheet and the remainder is folded along the dotted lines to make an open cuboid box, as shown in Figure 2.

1. Show that the volume, \(V \mathrm {~cm} ^ { 3 }\), of the box is given by $$V = 4 x ^ { 3 } - 80 x ^ { 2 } + 375 x$$
2. Use calculus to find the value of \(x\), to 3 significant figures, for which the volume of the box is a maximum.
3. Justify that this value of \(x\) gives a maximum value for \(V\).
4. Find, to 3 significant figures, the maximum volume of the box.
   \section*{8.} \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{6f9ace43-747b-419f-a9d1-d30165d77379-22_670_1004_292_392} \captionsetup{labelformat=empty} \caption{Figure 3}
   \end{figure} Figure 3 shows a sketch of the curve with equation \(y = \mathrm { f } ( x ) , x \in \mathbb { R }\). The curve crosses the \(y\)-axis at the point \(( 0,5 )\) and crosses the \(x\)-axis at the point \(( 6,0 )\). The curve has a minimum point at \(( 1,3 )\) and a maximum point at \(( 4,7 )\). On separate diagrams, sketch the curve with equation

1. \hspace{0pt} [In this question you may assume the formula for the area of a circle and the following formulae:
   a sphere of radius \(r\) has volume \(V = \frac { 4 } { 3 } \pi r ^ { 3 }\) and surface area \(S = 4 \pi r ^ { 2 }\) a cylinder of radius \(r\) and height \(h\) has volume \(V = \pi r ^ { 2 } h\) and curved surface area \(S = 2 \pi r h ]\)

\begin{figure}[h] \end{figure} Figure 5 shows the model for a building. The model is made up of three parts. The roof is modelled by the curved surface of a hemisphere of radius \(R \mathrm {~cm}\). The walls are modelled by the curved surface of a circular cylinder of radius \(R \mathrm {~cm}\) and height \(H \mathrm {~cm}\). The floor is modelled by a circular disc of radius \(R \mathrm {~cm}\). The model is made of material of negligible thickness, and the walls are perpendicular to the base. It is given that the volume of the model is \(800 \pi \mathrm {~cm} ^ { 3 }\) and that \(0 < R < 10.6\)

1. Show that $$H = \frac { 800 } { R ^ { 2 } } - \frac { 2 } { 3 } R$$
2. Show that the surface area, \(A \mathrm {~cm} ^ { 2 }\), of the model is given by $$A = \frac { 5 \pi R ^ { 2 } } { 3 } + \frac { 1600 \pi } { R }$$
3. Use calculus to find the value of \(R\), to 3 significant figures, for which \(A\) is a minimum.
4. Prove that this value of \(R\) gives a minimum value for \(A\).
5. Find, to 3 significant figures, the value of \(H\) which corresponds to this value for \(R\).

13. \begin{figure}[h] \end{figure} Figure 3 shows a flowerbed. Its shape is a quarter of a circle of radius \(x\) metres with two equal rectangles attached to it along its radii. Each rectangle has length equal to \(x\) metres and width equal to \(y\) metres. Given that the area of the flowerbed is \(4 \mathrm {~m} ^ { 2 }\),

1. show that $$y = \frac { 16 - \pi x ^ { 2 } } { 8 x }$$
2. Hence show that the perimeter \(P\) metres of the flowerbed is given by the equation $$P = \frac { 8 } { x } + 2 x$$
3. Use calculus to find the minimum value of \(P\).

1. In this question you must show all stages of your working.

\section*{Solutions based entirely on calculator technology are not acceptable.} \begin{figure}[h] \end{figure} A brick is in the shape of a cuboid with width \(x \mathrm {~cm}\) ,length \(3 x \mathrm {~cm}\) and height \(h \mathrm {~cm}\) ,as shown in Figure 2. The volume of the brick is \(972 \mathrm {~cm} ^ { 3 }\)

1. Show that the surface area of the brick,\(S \mathrm {~cm} ^ { 2 }\) ,is given by $$S = 6 x ^ { 2 } + \frac { 2592 } { x }$$
2. Find \(\frac { \mathrm { d } S } { \mathrm {~d} x }\)
3. Hence find the value of \(x\) for which \(S\) is stationary.
4. Find \(\frac { \mathrm { d } ^ { 2 } S } { \mathrm {~d} x ^ { 2 } }\) and hence show that the value of \(x\) found in part(c)gives the minimum value of \(S\) .
5. Hence find the minimum surface area of the brick.

1. Wheat is grown on a farm.

• In year 1 , the farm produced 300 tonnes of wheat.
• In year 12 , the farm is predicted to produce 4000 tonnes of wheat.

Model \(A\) assumes that the amount of wheat produced on the farm will increase by the same amount each year.

1. Using model \(A\), find the amount of wheat produced on the farm in year 4. Give your answer to the nearest 10 tonnes. Model \(B\) assumes that the amount of wheat produced on the farm will increase by the same percentage each year.
2. Using model \(B\), find the amount of wheat produced on the farm in year 2. Give your answer to the nearest 10 tonnes.
3. Calculate, according to the two models, the difference between the total amounts of wheat predicted to be produced on the farm from year 1 to year 12 inclusive. Give your answer to the nearest 10 tonnes.

1. The weight of a baby mammal is monitored over a 16 -month period.

The weight of the mammal, \(w \mathrm {~kg}\), is given by $$w = \log _ { a } ( t + 5 ) - \log _ { a } 4 \quad 2 \leqslant t \leqslant 18$$ where \(t\) is the age of the mammal in months and \(a\) is a constant.
Given that the weight of the mammal was 10 kg when \(t = 3\)

1. show that \(a = 1.072\) correct to 3 decimal places. Using \(a = 1.072\)
2. find an equation for \(t\) in terms of \(w\)
3. find the value of \(t\) when \(w = 15\), giving your answer to 3 significant figures.

1. A diesel lorry is driven from Birmingham to Bury at a steady speed of v kilometres per hour. The total cost of the journey, \(\pounds C\), is given by

$$C = \frac { 1400 } { v } + \frac { 2 v } { 7 } .$$

1. Find the value of \(v\) for which \(C\) is a minimum.
2. Find \(\frac { \mathrm { d } ^ { 2 } C } { \mathrm {~d} v ^ { 2 } }\) and hence verify that \(C\) is a minimum for this value of \(v\).
3. Calculate the minimum total cost of the journey.

10. A solid right circular cylinder has radius \(r \mathrm {~cm}\) and height \(h \mathrm {~cm}\). The total surface area of the cylinder is \(800 \mathrm {~cm} ^ { 2 }\).

1. Show that the volume, \(V \mathrm {~cm} ^ { 3 }\), of the cylinder is given by $$V = 400 r - \pi r ^ { 3 }$$ Given that \(r\) varies,
2. use calculus to find the maximum value of \(V\), to the nearest \(\mathrm { cm } ^ { 3 }\).
3. Justify that the value of \(V\) you have found is a maximum. \includegraphics[max width=\textwidth, alt={}, center]{12e54724-64a3-4dc0-b7d5-6ef6cc04124c-16_103_63_2477_1873}

8. \begin{figure}[h] \end{figure} A manufacturer produces pain relieving tablets. Each tablet is in the shape of a solid circular cylinder with base radius \(x \mathrm {~mm}\) and height \(h \mathrm {~mm}\), as shown in Figure 3. Given that the volume of each tablet has to be \(60 \mathrm {~mm} ^ { 3 }\),

1. express \(h\) in terms of \(x\),
2. show that the surface area, \(A \mathrm {~mm} ^ { 2 }\), of a tablet is given by \(A = 2 \pi x ^ { 2 } + \frac { 120 } { x }\) The manufacturer needs to minimise the surface area \(A \mathrm {~mm} ^ { 2 }\), of a tablet.
3. Use calculus to find the value of \(x\) for which \(A\) is a minimum.
4. Calculate the minimum value of \(A\), giving your answer to the nearest integer.
5. Show that this value of \(A\) is a minimum.

9. \begin{figure}[h] \end{figure} Figure 4 shows the plan of a pool. The shape of the pool \(A B C D E F A\) consists of a rectangle \(B C E F\) joined to an equilateral triangle \(B F A\) and a semi-circle \(C D E\), as shown in Figure 4. Given that \(A B = x\) metres, \(E F = y\) metres, and the area of the pool is \(50 \mathrm {~m} ^ { 2 }\),

1. show that $$y = \frac { 50 } { x } - \frac { x } { 8 } ( \pi + 2 \sqrt { } 3 )$$
2. Hence show that the perimeter, \(P\) metres, of the pool is given by $$P = \frac { 100 } { x } + \frac { x } { 4 } ( \pi + 8 - 2 \sqrt { } 3 )$$
3. Use calculus to find the minimum value of \(P\), giving your answer to 3 significant figures.
4. Justify, by further differentiation, that the value of \(P\) that you have found is a minimum.

10. \begin{figure}[h] \end{figure} \begin{figure}[h] \end{figure} Figure 4 shows a closed letter box \(A B F E H G C D\), which is made to be attached to a wall of a house. The letter box is a right prism of length \(y \mathrm {~cm}\) as shown in Figure 4. The base \(A B F E\) of the prism is a rectangle. The total surface area of the six faces of the prism is \(S \mathrm {~cm} ^ { 2 }\). The cross section \(A B C D\) of the letter box is a trapezium with edges of lengths \(D A = 9 x \mathrm {~cm}\), \(A B = 4 x \mathrm {~cm} , B C = 6 x \mathrm {~cm}\) and \(C D = 5 x \mathrm {~cm}\) as shown in Figure 5.
The angle \(D A B = 90 ^ { \circ }\) and the angle \(A B C = 90 ^ { \circ }\). The volume of the letter box is \(9600 \mathrm {~cm} ^ { 3 }\).

1. Show that $$y = \frac { 320 } { x ^ { 2 } }$$
2. Hence show that the surface area of the letter box, \(S \mathrm {~cm} ^ { 2 }\), is given by $$S = 60 x ^ { 2 } + \frac { 7680 } { x }$$
3. Use calculus to find the minimum value of \(S\).
4. Justify, by further differentiation, that the value of \(S\) you have found is a minimum.

8. A dose of antibiotics is given to a patient. The amount of the antibiotic, \(x\) milligrams, in the patient's bloodstream \(t\) hours after the dose was given, is found to satisfy the equation $$\log _ { 10 } x = 2.74 - 0.079 t$$

1. Show that this equation can be written in the form $$x = p q ^ { - t }$$ where \(p\) and \(q\) are constants to be found. Give the value of \(p\) to the nearest whole number and the value of \(q\) to 2 significant figures.
2. With reference to the equation in part (a), interpret the value of the constant \(p\). When a different dose of the antibiotic is given to another patient, the values of \(x\) and \(t\) satisfy the equation $$x = 400 \times 1.4 ^ { - t }$$
3. Use calculus to find, to 2 significant figures, the value of \(\frac { \mathrm { d } x } { \mathrm {~d} t }\) when \(t = 5\)

1. In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable.

A population of fruit flies is being studied.
The number of fruit flies, \(F\), in the population, \(t\) days after the start of the study, is modelled by the equation $$F = \frac { 350 \mathrm { e } ^ { k t } } { 9 + \mathrm { e } ^ { k t } }$$ where \(k\) is a constant.
Use the equation of the model to answer parts (a), (b) and (c).

1. Find the number of fruit flies in the population at the start of the study. Given that there are 200 fruit flies in the population 15 days after the start of the study,
2. show that \(k = \frac { 1 } { 15 } \ln 12\) Given also that, when \(t = T\), the number of fruit flies in the population is increasing at a rate of 10 per day,
3. find the possible values of \(T\), giving your answers to one decimal place.