Riddler Express, August 28 2020

I didn’t add last week’s Classic to this blog, but I did produce a pretty graph in Jupyter from which you can read the right answer (in either graph, the point on the x-axis where the orange four-post line first exceeds the blue three-post line.)

This week’s Express is not difficult, but I did find the answer a little surprising.

The question mentions (not explicitly) that the perimeter of a circle increases by 2π metres when the radius increases by one metre. Without thinking about it much, I expected the area to increase by a relatively small amount as well. The area of a sphere is $4\pi r^{2}$ , and the radius of the Earth, $r_{0}$ , is given as 6 378 000m.

change in area	=	new larger area – original area
	=	$4\pi (r{_{0}}+1)^{2}-4\pi (r{_{0}}^{2})$
	=	$4\pi ((r_{0}+1)^{2}-r_{0}^{2})$
	=	$4\pi(r_{0}^{2}+2r_{0}+1-r_{0}^{2})$
	=	$4\pi (2r_{0}+1)$
	=	$4\pi (2\times 6 378 000+1)$
	=	160296636.123… square metres

That’s about 160 square kilometres. According to Wikipedia, there’s no lake of similar size either in the UK or in Poland. It’s about one tenth the size of Greater London (1 569 square kilometres.)

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