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Monday, 31 March 2014

Sunday Afternoon Maths VI Answers & Extensions

This post contains the answers to this week's Sunday Afternoon Maths and some extension problems based around the originals.


The area of an ellipse is πab where a and b are the distances from the centre of the ellipse to the closest and furthest points on the ellipse.
In the first ellipse, a=5cm and b=4cm, so the area is 20πcm2. In the second ellipse, a=5cm and b=3cm, so the area is 15πcm2. Hence, the first ellipse has the larger area.
How far apart should the pins be placed to give the ellipse with the largest area?

Triangle Numbers

Tn = 1/2n(n+1), so:
Tn+Tn+1 = 1/2n(n+1) + 1/2(n+1)(n+2)
    = (n+1)2
So, we are looking for n such that (n+1)2+(n+3)2=(n+5)2. This is true when n=5 (62+82=102).
Find n such that Tn+Tn+1+Tn+1+Tn+2=Tn+2+Tn+3.

Ticking Clock

The second hand will always be pointing at one of the 60 graduations. If the minute and hour hand are 120° away from the second hand they must also be pointing at one of the graduations. The minute hand will only be pointing at a graduation at zero seconds past the minute, so the second hand must be pointing at 0. Therefore the hand are either pointing at: hour: 4, minute: 8, second: 0; or hour: 8, minute: 4, second: 0. Neither of these are real times, so it is not possible.
If the second hand moves continuously instead of moving every secone, will there be a time when the hands of the clock are all 120° apart?

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