The leaning ladder. (POW)
If this affects you, don’t read it. If you do, you can’t say I didn’t tell you not to…
A seven foot ladder rests against a wall and also rests against a cubical box ( length of side 1 foot ) which is sitting at the base of the wall and flush against it. If the ladder touches higher up the wall than it sticks out along the floor, where does teh ladder touch the wall?
When initially looking at that, I see/think:
- Three similar triangles given such that no sides correspond
- Three right triangles with missing sides
- The possibility of solving with inequalities by limiting the range of possible answers
- The frightening possibility of high degree polynomials ( like 5-6 )
- An opportunity to befriend coordinates
There being similar triangles, I used them. There being missing sides in right triangles, I used the pythagorean theorem. Let the height of the ladder on the wall be y and let x be the distance of the bottom of the ladder from the base of the wall. Clearly, x2+y2=72, thus y=sqrt{49-x2}. Setting up a proportion between the large triangle ( whole ladder as hypotenuse )and the right most triangle ( height of 1 ) gives: y/x=1/(x-1), therefore y=x/(x-1)>. The range of the possible answers can be greatly narrowed by noting that: the height has to be greater than it would be were the triangle isosceles and: the base must be greater than 1. Taking these inequalities into account, the interssection in this narrowed domain and range is the answer I obtained. This leaves my weekend free to find other, possibly eaiser, possibly more interesting ways to solve this.
I finally made a style changer button thinga. Its in the upper right. now, maybe I should finish starbucks..