## Sunday, May 24, 2009

### Inheritance

A rich land owner told his three sons that each will get a piece of land as big as he can walk around in one day. Oldest son decided to walk around a square contour, middle son around a circle and youngest son around a pentagon contour. Assuming that they all walked with the same speed and each returned to the point he started from, who got the largest piece of land?

rachel_t said...

The son who walked in a circle.

They all have the same perimeter, that I'll set as 1.

--Square--
each side is perimeter/4, or 1/4
Area = 0.25 x 0.25
Area = 0.0625

--Circle--
radius = 1/(2 x pi) = 0.159155
Area = pi x r^2 = pi x 0.159155^2
Area = 0.079577

--Pentagon--
I cheated and looked up the area...
each side has length of perimeter/5 or 1/5
if you break the pentagon into 10 triangles, the height of each triangle can be found by the tangent of the angle.
apothem (height)= (1/2)(1/5)/(tan (.5 x 360/5))
= 0.137638
Area = (1/2)(1/5)(0.137638)(5)
Area = 0.068819

Maria said...

Very impressive!
Rachel T. - we all should keep your contact information in mind when our time comes to divide the land.