Question

If an ice rink is `sqrt(404)` across then what is its perimeter?

Answer 1

It could be `44`, but I'm not particularly convinced.

Explanation:

Since there is not enough information in the question, let us work with the following assumptions:

• The ice rink is rectangular.

• The length of each side is an integer.

• The diagonal measurement of the ice rink is `sqrt(404)`.

• The rectangle is as close as possible to being square given the above conditions.

By Pythagoras, we can tell that we need a pair of positive integers, the sum of whose squares is `404`.

Note that:

`14 = sqrt(196) < sqrt(404/2) = sqrt(202) < sqrt(225) = 15`

So one of the sides of the rectangle is `>= 15` and the other `<= 14`.

Trying each possible larger side from `15` upwards, we find:

• `404 - 15^2 = 404-225 = color(red)(cancel(color(black)(179)))`

• `404 - 16^2 = 404-256 = color(red)(cancel(color(black)(148)))`

• `404 - 17^2 = 404-289 = color(red)(cancel(color(black)(115)))`

• `404 - 18^2 = 404-324 = color(red)(cancel(color(black)(80)))`

• `404 - 19^2 = 404-361 = color(red)(cancel(color(black)(43)))`

• `404 - 20^2 = 404-400 = 4 = 2^2`

So with the above assumptions, the ice rink is `20 xx 2`, with perimeter `20 + 2 + 20 + 2 = 44`.

Well there's an answer, but `20 xx 2` seems strange proportions for a real ice rink.