I wrote these in early 2013, having switched to Edexcel syllabi and in Beijing. This is written at a level for bright students from Year 10 to Year 13. Many of the questions call for some extended research and for the reader to do its own modelling. I could imagine setting these as holiday work, as in setting a challenge to do two questions, one from the top and one from the bottom. I would enjoy entering into extended interchange over any of these.

DJS 20130128 Email: david@scoins.net (what else ?)

1. How long does one stop add to a train journey? (a) Draw a velocity-time graph showing a train stopping within a longer journey. Assume 5 mins stopped, acceleration (as an absolute value) in the region of 0.05 m/s/s and a design speed of 120kph. (b) turn this into a formula, given speed v, stop S and acceleration a. (c) explore the difference in a journey of 500km with no stops and a top speed of 200kph and one with four stops. State any necessary assumptions.

2. Is it worth growing your own vegetables? What do you need to know? Example information; tomato seed is ...p per packet, so even a germination rate of 50% suggests that effectively the price is close to zero. Your time with the plants is in the region of 20 minutes per fruit total; minimum wage is GBP £ N /hour. Tomatoes cost between ... and ... depending on the time of year. Tomatoes grow outside need more attention; those grown under glass require capital expenditure, those grown inside the house need daily attention. None of the arithmetic is difficult, but you have to do it to discover answers. If you value your time doing this at the minimum wage, what is the minimum cost of tomatoes per kilogram?

3. Aside: tomatoes only just float, so their density is much the same as water. The volume of a tomato diameter d is close to d³/2; so how may 6cm tomatoes to the kilogramme ?

I have always assumed that the apple that landed on Newton weighed one Newton—how small does that make the apple?

4. (Q2 viewed more generally) Vegetable seed is cheap; they are easily grown, but they do require regular attention to permit free growth without weeds choking that growth. Assume seed cost is negligible, but that your time is valued at minimum wage. Assess how much time you need to grow any plant and what therefore it is worth to you. Assume that losses occur and that more time from you increases the proportion of the plant (e.g. its fruit) lasting to be eaten. Your land has cost, too; use £2000 per acre as a guideline. This calculation takes some research; when you have a result compare your figure with the price of purchase at a local shop. Compare fairly, but produce a figure that represents the value you attach to growing your own food. Choose an appropriate term from economics to represent this.

Supplementary matters: if you are paid n times more than minimum wage, how do you have to value your leisure time to make anything or grow anything? Is this an argument for doing nothing but more work? Of course it isn’t, but it might be an argument that says you ought to be the best you can at whatever it is that you do. If you find yourself tempted to employ others (or to buy the fruits of their labours, haha) you must, of course, cost your time in the selection of vendor, your own quality control, your inability to make good selections (perhaps from product ignorance) and you should put a cost to the wastage of resource resulting from those (in)activities. That will produce two effects; (i) you will (I hope) find ways of valuing your leisure time in a non-monetary way and (ii) where money is spent, you will have added value in those exchanges (e.g. I don’t want this trinket or junket but it is part of the price of this day out).

5. If a farmer can grow potatoes at peak yield of say 12 tonnes per hectare, and if a forty hectare growing area is as much as he can work on his own, what does that represent as a return on the use of the land and his time spent? What heavy equipment did you assume was necessary? You will need to research some appropriate figures. Find values of yield appropriate to your area's agriculture and apply the same thinking to a local crop. Expand your calculation to estimate the land you need to provide food for your country. Also, look at what happens if there is no machinery, working by hand; what is a reasonable area per individual to work? You might look at peasant farming to assess this. What then is the return on this? Can you then make an assessment about the support that first world countries can sensibly give to third world ones? This could easily form an extended essay, from GCSE to undergraduate level.

These next few questions require a grasp of differentiation, as for Further Pure Maths IGCSE and for C1 in Year 12.

Year 11 have shown recently that the cuboid box with the least area for a volume (and which is also the greatest volume for a given area) is a cube. So what is there about a box that makes the design so rarely a cube? Might the shape be affected by the amount of overlap in the design? Might the cost be governed by some factors other than material? Dare I suggest that the answers might earn you an interview with a packaging / storage company?

6. A folding box is made from a large rectangular sheet, which is cut, folded, stapled and taped. The finished box will have dimensions a,b,c; the sheet's shorter dimension is b+c, the longer dimension is 2a+2b. The cutting length is 8b, the fold length is 4(a+b+c). Assume cutting costs twice as much as folding, that folding costs ¥p/unit length; write down an expression for the cost of forming a box. Add one edge c, b<a, stapled at the same cost as two cuts. Leave the taping to the customer. You now have a formula with three variables.

To find some solutions, we need a second equation; we want, perhaps, a box that works well with our sheet; write an expression for the area of our sheet. Suppose our sheet is 8’x4’ or that shape, 2 units by 1. You now have two more equations from the given dimensions. Does that tell you the size of a and b?

Can you find the box with the biggest volume? What about the best value box (the least cost for the most volume)?

Did you *need* to work out what my design was to do the question?

7. A sector of a circle of perimeter P, radius r and angle θ is of unit area, say one square metre. The cost of making this is governed by the length of cutting, so find the θ that minimises the cutting; show that θ is between 90 and 115 degrees.

This sector is then bent to form a cone of height H and radius R. Find the volume of this cone. [Any difficulty here lies in you needing to find answers on the way to answering the question posed. The number of sub-stages, not the number of answers, is what determines the level of difficulty.]

8. For a similar cone to the last question, what would be the shape that gives maximum volume for fixed area? What would be the perimeter of this different sector before bending?

9. Assume a population, x, grows seasonally, one cycle per year. Let the growth rate be dx/dt = 5+2sin t. Without use of calculus, state the limits of rate of growth. By integration, find a formula for x in terms of t, given that at time t=0, x=3. You have assumed so far that no deaths occur; suppose the deaths, y, are also cyclical, at a rate that hits peaks twice a year, at extremes of temperature, such as dy/dt = k cos 2t. Write a condition that will give a stable population, hence find a value of k and find a formula for the live population at time t.

Do some research into the equation dx/dt = kx(1-x), which is the simplest form of the epidemic or sickness curve. Find more advanced models and write an essay to send to me on the topic.

DJS 201303

No answers for most of this, mostly because it ought to generate discussion and argument. Some of these questions would provide a whole week’s Y8 or Y9 maths lessons on their own.

6. Box Cost = material plus forming = material + cutting+folding+stapling

Forming = fold cost x 4(a+b+c) + 2xfold cost x 8c + 2x2b x fold cost = ¥(4a+4b+4c+16b+4c) =4¥(a+5b+2c).

The dimensions are 2(a+b)=2, so a+b=1, and b+c=1. We know no more, but the maximum volume will occur when a=b=c = 0.5 units. So that volume would be 0.125 cubic units.

To minimise our manufacturing cost, we would want a+5b+2c to be smallest, suggesting a:b:c = 10:2:5. However, if we use the dimension information to eliminate a&c, we have Cost=3(1+b) and Volume=b(1-b)². Differentiating both expressions and dividing one by the other gives a maximum at b=1/√3 and a=c=1-b, which suggests a cost of 12¥(1+1/√3) = 19¥ (ish) and a volume of 0.103 cubic units, dimensions roughly a=c=0.42, b=0.58.

One or both of a & b (but preferably a) is slightly reduced to accommodate the staple flange. This design has very little wasted sheet material, between 2 and 5%. My cutting from the rectangular sheet ran from the edges to the 8 corners, cutting out narrow triangles. The flange for stapling is around 0.05 of a unit, but in practice is between 2 and 5cm. In the real world all forming costs would count the number and the length (of cut, fold or staple), so my cutting would be 10 equal cuts of close to b/2, plus the two tiny cuts for the staple flange; the folding would be 2 of the longest dimension and five of length c (one is for the flange). So each of these costs would have a linear formula. If someone responds, I will write a whole page about boxes.