MEI Maths Item of the Month December 2013 Choosing crates with Santa

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MEI Maths Item of the Month
December 2013
Choosing crates with Santa
Santa is choosing which of two crates to use on his sled. They are both 5m high and both
have bases with diagonals 10m. One of them has a square base and the other has an
oblong base.
Santa wants to use the crate with the largest volume. Which one should he choose and
why?
Solution
As the two crates have the same heights the one with the largest area base will have the
largest volume.
By comparing the triangles obtained from half the areas:


half the square is obtained from triangles ADC and DBC;
half the oblong is obtained from the triangles ADC and AED.
AED and DBC are similar.
The longest side of DBC (DC) is longer than the side of the square and the longest side of
AED (AD) is shorter than the side of the square.
Therefore the area of DBC is greater than the area of AED and hence the square has larger
area than the oblong.
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