“John and Martha are measuring the girth of huge trees in an ancient forest. John has his tape around one tree measuring a circumference of 628cm. Martha is standing one metre away from the tree. How much of the tape can Martha see?”
So here’s my solution:
From basic trigonometry we know that the circumference (C) of a circle is defined by the expression C = 2 x π x r, where r is the radius of the circle. So having been given the circumference as 628 centimetres, we can calculate the radius (r) as 628 / (2 x π). ie 628 / (2 x 3.14), which gives a value of 100 centimetres for the radius (r).
Using my Oxford Study Mathematics Dictionary (what a find that was in a second-hand bookshop), I discovered that the angle between the tangent and a line to the centre of the circle is a right-angle (ie 90°).
Now we know the length of two sides of the drawn right-angle triangle, namely r centimetres (the radius of the tree) and r+100 centimetres (the dotted line of this triangle, made up of the radius of the tree and the distance the viewer is from the tree).
Knowing the length of two sides of a right-angle triangle we can calculate all sorts of things, in particular the angle between the two radius lines marked r. This angle, using the cosine rule for right-angle triangles, can be determined from the following:
cosine(angle) = length of the adjacent side / length of the hypotenuse
ie cosine(angle) = r / (r+100) = 100 / (100 + 100) =0.5
Now from basic trigonometry we know that the cosine of 60° is 0.5, so the angle is 60°.
Now 60° is one-sixth of the total number of degrees making up a circle (360°), so the length of that part of the circumference covered by the two radius lines marked r must be one-sixth of the total circumference of the tree. ie one-sixth of 628 centimetres. ie 628/6
BUT this is only the right hand-side of the tree! You have to repeat the same calculation for the left-hand side of the tree. It’s exactly the same result as the right-hand side, so the total length of the part of the circumference between the left and right tangents (ie the lines of sight of the viewer) is 2 x 628/6. ie 209.3 centimetres. This is the length of tape that Martha can see.
I have to admit this is all a rather verbose way of explaining it and it may prove to be incorrect. I have to wait until tomorrow, when the BBC publish the answer, to find out if I’ve wasted my time!
UPDATE – RESULT – Clever old me!
One-third of it, ie 209cm
Suppose the tree is a cylinder centred at O, radius R and Martha is at M. Its circumference is 628cm so its radius is 100cm. This means that Martha is a distance R from the surface of the cylinder. The tangent to the cylinder from M is at T and the angle MOT is 60 degrees. This means that Martha can see 120 degrees of the tape which is one third ie 209cm.”
things change 