Solution to Mensa Brain Bender Pipe Problem

 

Note from the reconstructed graphic the lime green (4 - x) triangle and the (y - 7) lavender triangle are similar.  Then:

4/x  =  y/ 7 ®  y = 28 x

Let L = the length of the pipe.  Then by the Pythagorean theorem:

L ²  = (4  + y) ²    +    (x + 7) ²

L ²  = (4 +   (28/ x)) ²    +    (x +x ² 7) ²

L²   = 16 + 224/x +  784/ x ² +   x ²    +  14 x   + 49

L²   = x ²    +  14 x + 65  + 224/x +  784/ x ²

Now take the derivative of  L²:

( L²)' =  [x ²    +  14 x + 65  + 224/x +  784/ x ² ]'

( L²)' =  2x   +  14 + 0  - 224/x ²  - 1568 / x³

Set the derivative equal to 0 and solve for x:

0  =  2x   +  14  - 224/x ²  - 1568 / x³

Multiply both sides by  x³:

0  =  2x⁴   +  14x³  - 224x  - 1568

0  =  x³ (2x + 14) - 112(2x + 14)

0  =  (x³ - 112) (2x + 14)

x =  ³Ö 112,   - 7

For x = - 7:

L²   = (-7) ²    +  14 (-7) + 65  + 224/(-7) +  784/ (-7) ²

=  49 - 98  +65 - 32 + 16 = 0  (invalid soln.)

For x = ³Ö 112 = 2 Ö 14

A pipe that's less than 15.36 ft. long will make it around the corner. (It is too short to touch both exterior walls and the corner at the intersection.)