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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.

#### Ellipses

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.
##### Extension
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).
##### Extension
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.
##### Extension
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?