Here’s a little bit of PHP that helps you to solve the Crocodile maths question.

### Problem

A crocodile is stalking prey located 20 metres further upstream on the opposite bank of a river.

Crocodiles travel at different speeds in land and in water.

The time taken for the crocodile to reach its prey can be minimised if it swims to a particular point, **P**, *x* metres upstream on the other side of the river as shown in the diagram.

The time taken, *T*, measured in tenths of a second, is given by

T(x) = 5 x sqrt( 36 + x squared ) + 4 ( 20 - x )

- a. i
- Calculate the time taken if the crocodile does not travel on land.
- a. ii
- Calculate the time taken if the crocodile swims the shortest distance possible.
- b.
- Between these two extremes there is one value
*x*which minimises the time taken. Find this value of*x*and calculate the minimum possible time.

[bw_api] $inc = 1; $mint = 999; for ( $x = 0; $x <= 20; $x += $inc ) { $t = ( 5 * sqrt( $x*$x + 36 )) + (4 * ( 20 - $x )); echo number_format( $x, 1) . " " . number_format( $t, 2 ). PHP_EOL; if ( $t < $mint ) { $mint = $t; $minx = $x; } } echo "Minimum t: $mint" . PHP_EOL; echo "for x: $minx" . PHP_EOL; [/bw_api] Output [bw_geshi none] C:\apache\htdocs\wordpress\wp-content\plugins\play>php croc.php 0.0 110.00 1.0 106.41 2.0 103.62 3.0 101.54 4.0 100.06 5.0 99.05 6.0 98.43 7.0 98.10 8.0 98.00 9.0 98.08 10.0 98.31 11.0 98.65 12.0 99.08 13.0 99.59 14.0 100.16 15.0 100.78 16.0 101.44 17.0 102.14 18.0 102.87 19.0 103.62 20.0 104.40 Minimum t: 98 for x: 8 [/bw_geshi]

### Answers

- a. i
- If the crocodile does not travel on land then it swims all the way.
`x=20`

so`t=104.4`

- a. ii
- The shortest distance to swim is straight across.
`x=0`

so`t=110.0`

- b
- The minimum value for t is
`98.0`

– when`x=8`

.

### Comments

- In this problem we ignore negative values for the square root
- We assume that either the stream is not flowing or the crocodile’s speed in water has been taken into account in the formula
- We assume the zebra is too stupid to run away
- I do hope I’ve got the answers right

### See also

- http://www.bbc.co.uk/news/uk-scotland-34476699
- http://www.telegraph.co.uk/education/educationnews/11921671/Can-you-answer-the-tricky-crocodile-maths-question-that-left-students-in-tears.html
- Had I used the right Jetpack module (Beautiful math) I could have shown the maths formula prettily; by writing the formula using LaTeX.

BTW: I’ve not watched this video yet

#### A few minutes later…

OK, I’ve watched the video now.

- My answers are in tenths of a second
- I didn’t do the maths
- I just got a computer to do the sums
- In my opinion the program was easier than the maths