1.02z Models in context: use functions in modelling

1. A research engineer is testing the effectiveness of the braking system of a car when it is driven in wet conditions.

The engineer measures and records the braking distance, \(d\) metres, when the brakes are applied from a speed of \(V \mathrm { kmh } ^ { - 1 }\). Graphs of \(d\) against \(V\) and \(\log _ { 10 } d\) against \(\log _ { 10 } V\) were plotted.
The results are shown below together with a data point from each graph. \begin{figure}[h] \end{figure} \begin{figure}[h] \end{figure}

1. Explain how Figure 6 would lead the engineer to believe that the braking distance should be modelled by the formula $$d = k V ^ { n } \quad \text { where } k \text { and } n \text { are constants }$$ with \(k \approx 0.017\) Using the information given in Figure 5, with \(k = 0.017\)
2. find a complete equation for the model giving the value of \(n\) to 3 significant figures. Sean is driving this car at \(60 \mathrm { kmh } ^ { - 1 }\) in wet conditions when he notices a large puddle in the road 100 m ahead. It takes him 0.8 seconds to react before applying the brakes.
3. Use your formula to find out if Sean will be able to stop before reaching the puddle.

13. \begin{figure}[h] \end{figure} [A sphere of radius \(r\) has volume \(\frac { 4 } { 3 } \pi r ^ { 3 }\) and surface area \(4 \pi r ^ { 2 }\) ]
A manufacturer produces a storage tank.
The tank is modelled in the shape of a hollow circular cylinder closed at one end with a hemispherical shell at the other end as shown in Figure 9. The walls of the tank are assumed to have negligible thickness.
The cylinder has radius \(r\) metres and height \(h\) metres and the hemisphere has radius \(r\) metres.
The volume of the tank is \(6 \mathrm {~m} ^ { 3 }\).

1. Show that, according to the model, the surface area of the tank, in \(\mathrm { m } ^ { 2 }\), is given by $$\frac { 12 } { r } + \frac { 5 } { 3 } \pi r ^ { 2 }$$ The manufacturer needs to minimise the surface area of the tank.
2. Use calculus to find the radius of the tank for which the surface area is a minimum.
   (4)
3. Calculate the minimum surface area of the tank, giving your answer to the nearest integer.

1. A quantity of ethanol was heated until it reached boiling point.

The temperature of the ethanol, \(\theta ^ { \circ } \mathrm { C }\), at time \(t\) seconds after heating began, is modelled by the equation $$\theta = A - B \mathrm { e } ^ { - 0.07 t }$$ where \(A\) and \(B\) are positive constants.
Given that

• the initial temperature of the ethanol was \(18 ^ { \circ } \mathrm { C }\)
• after 10 seconds the temperature of the ethanol was \(44 ^ { \circ } \mathrm { C }\)
  1. find a complete equation for the model, giving the values of \(A\) and \(B\) to 3 significant figures.

Ethanol has a boiling point of approximately \(78 ^ { \circ } \mathrm { C }\)

Use this information to evaluate the model.

5 The fuel consumption of a car, \(C\) miles per gallon, varies with the speed, \(v\) miles per hour. Jamal models the fuel consumption of his car by the formula \(C = \frac { 12 } { 5 } v - \frac { 3 } { 125 } v ^ { 2 }\), for \(0 \leqslant v \leqslant 80\).

1. Suggest a reason why Jamal has included an upper limit in his model.
2. Determine the speed that gives the maximum fuel consumption. Amaya's car does more miles per gallon than Jamal's car. She proposes to model the fuel consumption of her car using a formula of the form \(C = \frac { 12 } { 5 } v - \frac { 3 } { 125 } v ^ { 2 } + k\), for \(0 \leqslant v \leqslant 80\), where \(k\) is a positive constant.
3. Give a reason why this model is not suitable.
4. Suggest a different change to Jamal's formula which would give a more suitable model.

11 David puts a block of ice into a cool-box. He wishes to model the mass \(m \mathrm {~kg}\) of the remaining block of ice at time \(t\) hours later. He finds that when \(t = 5 , m = 2.1\), and when \(t = 50 , m = 0.21\).

1. David at first guesses that the mass may be inversely proportional to time. Show that this model fits his measurements.
2. Explain why this model
   1. is not suitable for small values of \(t\),
   2. cannot be used to find the time for the block to melt completely. David instead proposes a linear model \(m = a t + b\), where \(a\) and \(b\) are constants.
3. Find the values of the constants for which the model fits the mass of the block when \(t = 5\) and \(t = 50\).
4. Interpret these values of \(a\) and \(b\).
5. Find the time according to this model for the block of ice to melt completely.

8 A team of volunteers donates cakes for sale at a charity stall. The number of cakes that can be sold depends on the price. A model for this is \(\mathrm { y } = 190 - 70 \mathrm { x }\), where \(y\) cakes can be sold when the price of a cake is \(\pounds\) x.

1. Find how many cakes could be given away for free according to this model. The number of volunteers who are willing to donate cakes goes up as the price goes up. If the cakes sell for \(\pounds 1.20\) they will donate 50 cakes, but if they sell for \(\pounds 2.40\) they will donate 140 cakes. They use the linear model \(\mathrm { y } = \mathrm { mx } + \mathrm { c }\) to relate the number of cakes donated, \(y\), to the price of a cake, \(\pounds x\).
2. Find the values of the constants \(m\) and \(c\) for which this linear model fits the two data points.
3. Explain why the model is not suitable for very low prices.
4. The team would like to sell all the cakes that they donate. Find the set of possible prices that the cakes could have to achieve this.

10 Zac is measuring the growth of a culture of bacteria in a laboratory. The initial area of the culture is \(8 \mathrm {~cm} ^ { 2 }\). The area one day later is \(8.8 \mathrm {~cm} ^ { 2 }\). At first, Zac uses a model of the form \(\mathrm { A } = \mathrm { a } + \mathrm { bt }\), where \(A \mathrm {~cm} ^ { 2 }\) is the area \(t\) days after he begins measuring and \(a\) and \(b\) are constants.

1. Find the values of \(a\) and \(b\) that best model the initial area and the area one day later.
2. Calculate the value of \(t\) for which the model predicts an area of \(15 \mathrm {~cm} ^ { 2 }\).
3. Zac notices the area covered by the culture increases by \(10 \%\) each day. Explain why this model may not be suitable after the first day. Zac decides to use a different model for \(A\). His new model is \(\mathrm { A } = \mathrm { Pe } ^ { \mathrm { kt } }\), where \(P\) and \(k\) are constants.
4. Find the values of \(P\) and \(k\) that best model the initial area and the area one day later.
5. Calculate the value of \(t\) for which the area reaches \(15 \mathrm {~cm} ^ { 2 }\) according to this model.
6. Explain why this model may not be suitable for large values of \(t\).

14 Douglas wants to construct a model for the height of the tide in Liverpool during the day, using a cosine graph to represent the way the height changes. He knows that the first high tide of the day measures 8.55 m and the first low tide of the day measures 1.75 m . Douglas uses \(t\) for time and \(h\) for the height of the tide in metres. With his graph-drawing software set to degrees, he begins by drawing the graph of \(\mathrm { h } = 5.15 + 3.4\) cost.

1. Verify that this equation gives the correct values of \(h\) for the high and low tide. Douglas also knows that the first high tide of the day occurs at 1 am and the first low tide occurs at 7.20 am. He wants \(t\) to represent the time in hours after midnight, so he modifies his equation to \(h = 5.15 + 3.4 \cos ( a t + b )\).
2. 1. Show that Douglas's modified equation gives the first high tide of the day occurring at the correct time if \(\mathrm { a } + \mathrm { b } = 0\).
   2. Use the time of the first low tide of the day to form a second equation relating \(a\) and \(b\).
   3. Hence show that \(a = 28.42\) correct to 2 decimal places.
3. Douglas can only sail his boat when the height of the tide is at least 3 m . Use the model to predict the range of times that morning when he cannot sail.
4. The next high tide occurs at 12.59 pm when the height of the tide is 8.91 m . Comment on the suitability of Douglas's model.

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 \(t\).

7 In this question take \(\boldsymbol { g } = \mathbf { 1 0 }\).
A small stone is projected from a point O with a speed of \(26 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle \(\theta\) above the horizontal. The initial velocity and part of the path of the stone are shown in Fig. 7.
You are given that \(\sin \theta = \frac { 12 } { 13 }\).
After \(t\) seconds the horizontal displacement of the stone from O is \(x\) metres and the vertical displacement is \(y\) metres. \begin{figure}[h] \end{figure}

1. Using the standard model for projectile motion,
   The stone passes through a point A . Point A is 16 m above the level of O .
2. Find the two possible horizontal distances of A from O . A toy balloon is projected from O with the same initial velocity as the small stone.
3. Suggest two ways in which the standard model could be adapted.

10 In a certain region, the populations of grey squirrels, \(P _ { \mathrm { G } }\) and red squirrels \(P _ { \mathrm { R } }\), at time \(t\) years are modelled by the equations: \(P _ { \mathrm { G } } = 10000 \left( 1 - \mathrm { e } ^ { - k t } \right)\) \(P _ { \mathrm { R } } = 20000 \mathrm { e } ^ { - k t }\) where \(t \geq 0\) and \(k\) is a positive constant.

1. 1. On the axes in your Printed Answer Book, sketch the graphs of \(P _ { \mathrm { G } }\) and \(P _ { \mathrm { R } }\) on the same axes.
   2. Give the equations of any asymptotes.
2. What does the model predict about the long term population of
   Grey squirrels and red squirrels compete for food and space. Grey squirrels are larger and more successful than red squirrels.
3. Comment on the validity of the model given by the equations, giving a reason for your answer.
4. Show that, according to the model, the rate of decrease of the population of red squirrels is always double the rate of increase of the population of grey squirrels.
5. When \(t = 3\), the numbers of grey and red squirrels are equal. Find the value of \(k\).

16 In the first year of a course, an A-level student, Aaishah, has a mathematics test each week. The night before each test she revises for \(t\) hours. Over the course of the year she realises that her percentage mark for a test, \(p\), may be modelled by the following formula, where \(A , B\) and \(C\) are constants. $$p = A - B ( t - C ) ^ { 2 }$$

• Aaishah finds that, however much she revises, her maximum mark is achieved when she does 2 hours revision. This maximum mark is 62 .
• Aaishah had a mark of 22 when she didn't spend any time revising.
  1. Find the values of \(A , B\) and \(C\).
  2. According to the model, if Aaishah revises for 45 minutes on the night before the test, what mark will she achieve?
  3. What is the maximum amount of time that Aaishah could have spent revising for the model to work?

In an attempt to improve her marks Aaishah now works through problems for a total of \(t\) hours over the three nights before the test. After taking a number of tests, she proposes the following new formula for \(p\). $$p = 22 + 68 \left( 1 - \mathrm { e } ^ { - 0.8 t } \right)$$ For the next three tests she recorded the data in Fig. 16. \begin{table}[h] \end{table}

Verify that the data is consistent with the new formula.

Aaishah's tutor advises her to spend a minimum of twelve hours working through problems in future. Determine whether or not this is good advice.

5 A social media website launched on 1 January 2017. The owners of the website report the number of users the site has at the start of each month. They believe that the relationship between the number of users, \(n\), and the number of months after launch, \(t\), can be modelled by \(n = a \times 2 ^ { k t }\) where \(a\) and \(k\) are constants.

1. Show that, according to the model, the graph of \(\log _ { 10 } n\) against \(t\) is a straight line.
2. Fig. 5 shows a plot of the values of \(t\) and \(\log _ { 10 } n\) for the first seven months. The point at \(t = 1\) is for 1 February 2017, and so on. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{31bc8bde-8d37-4e97-94e2-e3e73aab55e9-6_831_1442_609_388} \captionsetup{labelformat=empty} \caption{Fig. 5}
   \end{figure} Find estimates of the values of \(a\) and \(k\).
3. The owners of the website wanted to know the date on which they would report that the website had half a million users. Use the model to estimate this date.
4. Give a reason why the model may not be appropriate for large values of \(t\).

4 A hollow cone is fixed with its axis vertical and its vertex downwards. A small sphere \(P\) of mass \(m \mathrm {~kg}\) is moving in a horizontal circle on the inner surface of the cone. An identical sphere \(Q\) rests in equilibrium inside the cone (see diagram). \includegraphics[max width=\textwidth, alt={}, center]{c6445493-9802-46ca-b7eb-7738a831d9ee-3_586_611_404_246} The following modelling assumptions are made.

• \(P\) and \(Q\) are modelled as particles.
• The cone is modelled as smooth.
• There is no air resistance.
  1. Assuming that \(P\) moves with a constant speed \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), show that the total mechanical energy of \(P\) is \(\frac { 3 } { 2 } \mathrm { mv } ^ { 2 } \mathrm {~J}\) more than the total mechanical energy of \(Q\).
  2. Explain how the assumption that \(P\) and \(Q\) are both particles has been used.

In practice, \(P\) will not move indefinitely in a perfectly circular path, but will actually follow an approximately spiral path on the inside surface of the cone until eventually it collides with \(Q\).

Suggest an improvement that could be made to the model.

6 The diagram shows a block of wood in the shape of a prism with triangular cross-section. The end faces are right-angled triangles with sides of lengths \(3 x \mathrm {~cm}\), \(4 x \mathrm {~cm}\) and \(5 x \mathrm {~cm}\), and the length of the prism is \(y \mathrm {~cm}\), as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{66813123-3876-4484-aad1-4bfc09bb1508-7_394_825_459_548} The total surface area of the five faces is \(144 \mathrm {~cm} ^ { 2 }\).

1. 1. Show that \(x y + x ^ { 2 } = 12\).
   2. Hence show that the volume of the block, \(V \mathrm {~cm} ^ { 3 }\), is given by $$V = 72 x - 6 x ^ { 3 }$$
2. 1. Find \(\frac { \mathrm { d } V } { \mathrm {~d} x }\).
   2. Show that \(V\) has a stationary value when \(x = 2\).
3. Find \(\frac { \mathrm { d } ^ { 2 } V } { \mathrm {~d} x ^ { 2 } }\) and hence determine whether \(V\) has a maximum value or a minimum value when \(x = 2\).
   (2 marks)

4 The diagram shows a solid cuboid with sides of lengths \(x \mathrm {~cm} , 3 x \mathrm {~cm}\) and \(y \mathrm {~cm}\). \includegraphics[max width=\textwidth, alt={}, center]{dbc25177-4a28-480f-93d5-41acb2a2d28c-3_349_472_376_769} The total surface area of the cuboid is \(32 \mathrm {~cm} ^ { 2 }\).

1. 1. Show that \(3 x ^ { 2 } + 4 x y = 16\).
   2. Hence show that the volume, \(V \mathrm {~cm} ^ { 3 }\), of the cuboid is given by $$V = 12 x - \frac { 9 x ^ { 3 } } { 4 }$$
2. Find \(\frac { \mathrm { d } V } { \mathrm {~d} x }\).
3. 1. Verify that a stationary value of \(V\) occurs when \(x = \frac { 4 } { 3 }\).
   2. Find \(\frac { \mathrm { d } ^ { 2 } V } { \mathrm {~d} x ^ { 2 } }\) and hence determine whether \(V\) has a maximum value or a minimum value when \(x = \frac { 4 } { 3 }\).

6 The diagram shows a cylindrical container of radius \(r \mathrm {~cm}\) and height \(h \mathrm {~cm}\). The container has an open top and a circular base. \includegraphics[max width=\textwidth, alt={}, center]{c7f38f7e-75aa-4b41-96fd-f38f968c225c-12_389_426_404_751} The external surface area of the container's curved surface and base is \(48 \pi \mathrm {~cm} ^ { 2 }\).
When the radius of the base is \(r \mathrm {~cm}\), the volume of the container is \(V \mathrm {~cm} ^ { 3 }\).

1. 1. Find an expression for \(h\) in terms of \(r\).
   2. Show that \(V = 24 \pi r - \frac { \pi } { 2 } r ^ { 3 }\).
2. 1. Find \(\frac { \mathrm { d } V } { \mathrm {~d} r }\).
   2. Find the positive value of \(r\) for which \(V\) is stationary, and determine whether this stationary value is a maximum value or a minimum value.
      [0pt] [4 marks]

7. Initially the number of fish in a lake is 500000 . The population is then modelled by the recurrence relation $$u _ { n + 1 } = 1.05 u _ { n } - d , \quad u _ { 0 } = 500000 .$$ In this relation \(u _ { n }\) is the number of fish in the lake after \(n\) years and \(d\) is the number of fish which are caught each year. Given that \(d = 15000\),

1. calculate \(u _ { 1 } , u _ { 2 }\) and \(u _ { 3 }\) and comment briefly on your results. Given that \(d = 100000\),
2. show that the population of fish dies out during the sixth year.
3. Find the value of \(d\) which would leave the population each year unchanged.

6. A container made from thin metal is in the shape of a right circular cylinder with height \(h \mathrm {~cm}\) and base radius \(r \mathrm {~cm}\). The container has no lid. When full of water, the container holds \(500 \mathrm {~cm} ^ { 3 }\) of water.

1. Show that the exterior surface area, \(A \mathrm {~cm} ^ { 2 }\), of the container is given by \(A = \pi r ^ { 2 } + \frac { 1000 } { r }\).
2. Find the value of \(r\) for which \(A\) is a minimum.
3. Prove that this value of \(r\) gives a minimum value of \(A\).
4. Calculate the minimum value of \(A\), giving your answer to the nearest integer.

3. \begin{figure}[h] \end{figure} Figure 2 shows the curve with equation \(y = \frac { 1 } { x ^ { 2 } + 1 }\).
The shaded region \(R\) is bounded by the curve, the coordinate axes and the line \(x = 2\).

1. Use the trapezium rule with four strips of equal width to estimate the area of \(R\). The cross-section of a support for a bookshelf is modelled by \(R\) with 1 unit on each axis representing 8 cm . Given that the support is 2 cm thick,
2. find an estimate for the volume of the support.

4 A ring is moving on a straight wire. Its velocity is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at time \(t\) seconds after passing a point Q .
Model A for the motion of the ring gives the velocity-time graph for \(0 \leqslant t \leqslant 6\) shown in Fig. 7 . \begin{figure}[h] \end{figure} Use model A to calculate the following.

1. The acceleration of the ring when \(t = 0.5\).
2. The displacement of the ring from Q when
   (A) \(t = 2\),
   (B) \(t = 6\). In an alternative model B , the velocity of the ring is given by \(v = 2 t ^ { 2 } - 14 t + 20\) for \(0 \leqslant t \leqslant 6\).
3. Calculate the acceleration of the ring at \(t = 0.5\) as given by model B.
4. Calculate by how much the models differ in their values for the least \(v\) in the time interval \(0 \leqslant t \leqslant 6\).
5. Calculate the displacement of the ring from Q when \(t = 6\) as given by model B .

2 The speed of a 100 metre runner in \(\mathrm { m } \mathrm { s } ^ { - 1 }\) is measured electronically every 4 seconds.
The measurements are plotted as points on the speed-time graph in Fig. 6. The vertical dotted line is drawn through the runner's finishing time. Fig. 6 also illustrates Model P in which the points are joined by straight lines. \begin{figure}[h] \end{figure}

1. Use Model P to estimate
   (A) the distance the runner has gone at the end of 12 seconds,
   (B) how long the runner took to complete 100 m . A mathematician proposes Model Q in which the runner's speed, \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at time \(t \mathrm {~s}\), is given by $$v = \frac { 5 } { 2 } t - \frac { 1 } { 8 } t ^ { 2 } .$$
2. Verify that Model Q gives the correct speed for \(t = 8\).
3. Use Model Q to estimate the distance the runner has gone at the end of 12 seconds.
4. The runner was timed at 11.35 seconds for the 100 m . Which model places the runner closer to the finishing line at this time? In this question take \(g\) as \(10 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
   A small ball is released from rest. It falls for 2 seconds and is then brought to rest over the next 5 seconds. This motion is modelled in the speed-time graph Fig. 6. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{4f80ea36-001f-4a00-849f-542f5072516b-3_658_1101_281_503} \captionsetup{labelformat=empty} \caption{Fig. 6}
   \end{figure} For this model,
5. calculate the distance fallen from \(t = 0\) to \(t = 7\),
6. find the acceleration of the ball from \(t = 2\) to \(t = 6\), specifying the direction,
7. obtain an expression in terms of \(t\) for the downward speed of the ball from \(t = 2\) to \(t = 6\),
8. state the assumption that has been made about the resistance to motion from \(t = 0\) to \(t = 2\). The part of the motion from \(t = 2\) to \(t = 7\) is now modelled by \(v = - \frac { 3 } { 2 } t ^ { 2 } + \frac { 19 } { 2 } t + 7\).
9. Verify that \(v\) agrees with the values given in Fig, 6 at \(t = 2 , t = 6\) and \(t = 7\).
10. Calculate the distance fallen from \(t = 2\) to \(t = 7\) according to this model.

2 Robin is driving a car of mass 800 kg along a straight horizontal road at a speed of \(40 \mathrm {~ms} ^ { - 1 }\).
Robin applies the brakes and the car decelerates uniformly; it comes to rest after travelling a distance of 125 m .

1. Show that the resistance force on the car when the brakes are applied is 5120 N .
2. Find the time the car takes to come to rest. For the rest of this question, assume that when Robin applies the brakes there is a constant resistance force of 5120 N on the car. The car returns to its speed of \(40 \mathrm {~ms} ^ { - 1 }\) and the road remains straight and horizontal.
   Robin sees a red light 155 m ahead, takes a short time to react and then applies the brakes.
   The car comes to rest before it reaches the red light.
3. Show that Robin's reaction time is less than 0.75 s . The 'stopping distance' is the total distance travelled while a driver reacts and then applies the brakes to bring the car to rest. For the rest of this question, assume that Robin is still driving the car described above and has a reaction time of 0.675 s . (This is the figure used in calculating the stopping distances given in the Highway Code.)
4. Calculate the stopping distance when Robin is driving at \(20 \mathrm {~ms} ^ { - 1 }\) on a horizontal road. The car then travels down a hill which has a slope of \(5 ^ { \circ }\) to the horizontal.
5. Find the stopping distance when Robin is driving at \(20 \mathrm {~ms} ^ { - 1 }\) down this hill.
6. By what percentage is the stopping distance increased by the fact that the car is going down the hill? Give your answer to the nearest 1\%.

1. A physics student is studying the movement of particles in an electric field. In one experiment, the distances in micrometres of two moving particles, \(A\) and \(B\), from a fixed point \(O\) are modelled by

$$\begin{aligned} & d _ { A } = | 5 t - 31 | \\ & d _ { B } = \left| 3 t ^ { 2 } - 25 t + 8 \right| \end{aligned}$$ respectively, where \(t\) is the time in seconds after motion begins.

1. Use algebra to find the range of time for which particle \(A\) is further away from \(O\) than particle \(B\) is from \(O\). It was recorded that the distance of particle \(B\) from \(O\) was less than the distance of particle \(A\) from \(O\) for approximately 4 seconds.
2. Use this information to assess the validity of the model.

8 \includegraphics[max width=\textwidth, alt={}, center]{e3942549-bfc0-432a-bf49-7d01d44af01a-6_533_524_246_772} The diagram shows a container which consists of a cylinder with a solid base and a hemispherical top. The radius of the cylinder is \(r \mathrm {~cm}\) and the height is \(h \mathrm {~cm}\). The container is to be made of thin plastic. The volume of the container is \(45 \pi \mathrm {~cm} ^ { 3 }\).

1. Show that the surface area of the container, \(A \mathrm {~cm} ^ { 2 }\), is given by $$A = \frac { 5 } { 3 } \pi r ^ { 2 } + \frac { 90 \pi } { r } .$$ [The volume of a sphere is \(V = \frac { 4 } { 3 } \pi r ^ { 3 }\) and the surface area of a sphere is \(S = 4 \pi r ^ { 2 }\).]
2. Use calculus to find the minimum surface area of the container, justifying that it is a minimum.
3. Suggest a reason why the manufacturer would wish to minimise the surface area.