This page contains some supplementary problems to be done alongside the normal work in your first term. There will be one per week when it is finished.
1.(a) What is the area of largest triangle that can be fitted inside a square of side 1? Is the triangle unique, and if not, can you classify them? (Only GCSE maths required! The point of this question is that you must produce a careful argument to show that your candidate largest triangle(s) is/are in fact the best possible. That is, you must eliminate all other possibilities.)
(b) What is the area of the smallest triangle that contains a square of side 1? Is the triangle unique?
2. While constructing a spreadsheet to record exam marks, I needed to solve the following problems. Given that the computer can add, subtract etc in the usual way, and that it has a built-in function max(.,.,.) which returns the maximum of any number of inputs, write a one-line formula to find each of:
(a) The minimum of a set of numbers.
(b) The second highest of a set of 4 numbers.
(The formula needs to be one-line in order to work from a cell in the spreadsheet. It would be much better to write a short algorithm to do (b): how would you do this?)
3. The image below (from NASA) shows the position of Mars in the sky on a sequence of consecutive days in 2005. It starts in the West, at the bottom right, and moves to the East, top left (the picture was taken with an astronomical telescope so it is upside down).
You see that for several days, Mars moves 'backewards': this is called retrograde motion. Explain why this should occur, using a simple cartoon of the solar system in which the plantes orbit the sun in circles. (You should write down the position vectors of Earth and Mars in parametric form (with time t as the parameter) and then you can calculate conditions for retrograde motion to occur.)
While you are at the NASA site, take a look at some of their other images.
4. Take the z axis vertically upwards. The curve x = r(z) is rotated about this axis to form a surface of revolution. Draw this surface for the cases (a) r(z) = z, (b) r(z) = z2 ; (c) r(z) = z½ ; (d) r(z) = zn , n = 3, 4, ...; (e) r(x) = cosh z.. . Explain how to calculate the volume of this surface of revolution between two points on the z axis.
A bowl is made in the shape of a surface of rotation generated by r(z). It is filled with water to a depth z0. Starting from time t = 0, the water drains out through a small hole at the origin in such a way that the volume flux is proportional to the depth. Why is this a plausible model? Calculate the depth of water z(t) as a function of time for a circular cylindrical bowl r(z) = 1.
Suppose that the water level z(t) decreases at a constant rate. What shape is the bowl?
5. Prove the well known result that (1 + z/n)n → ez as n tends to infinity (full rigour not required; for example, take logs of both sides and use the Taylor expansion of log(1 + a) for small a.)
Draw the curves y = xn for 0 < x < 2 and a range of inceasing n. Now consider the limiting behaviour as n tends to infinity. First fix x < 1. What is the limit? Repeat for fixed x > 1. Lastly put x = 1 + z/n. for fixed z. What happens now? Use these pieces of information to describe the limiting behaviour fully.
Repeat for the curves y = x1/n (no extra work required!).
6. These pictures show Lough Hyne (the 'H' is silent, and Lough means Lake) in Ireland. No river flows winto the Lough, but it is connected to the sea by a narrow channel a few hundred metres long and several metres wide. For 8 hours in every 12, water flows rapidly through this channel; for 4 hours, it flows in the other direction; there are 15-minute slack periods in between (see the right-hand picture). Explain this phenomenon, to include a prediction of which way the water flows for 8 hours and which for 4.