A field in the shape of a right triangle (with the two shortest sides measuring 60 feet and 80 feet) has roads on all three sides that don't drain properly. As a result, muddy water puddles collect and when the cars pass through, they splash all the snow that is within 5 feet of any one of the three roads.

If all other snow not splashed by cars is kept clean, what percent is clean?

The inradius can be employed to great advantage here. The inradius of the big triangle is (a+b-c)/2=20.  The inradius of the similar little triangle inside the big triangle is thus 20-5=15. The linear proportionality factor is thus 15/20=3/4.  The area ratio is thus (3/4)²=9/16 or 56.25%.

The formula (a+b-c)/2 for the inradius r is easily deduced based on calculating the area of the right triangle in two ways: as half of base times height, or as the sum of the areas of the square and the two "kites" at the vertices that are formed using the radii of the incircle from its points of tangency with the triangle. Thus ab/2= r(a-r)+r(b-r)+r² and solving the quadratic gives r=(a+b+c)/2 and r=(a+b-c)/2. Only the smaller root can be the radius of the inscribed circle, of course.

Edited on May 16, 2005, 12:43 am

Posted by Richard on 2005-05-16 00:42:13