Claudette Tucker

Problem: Ladder and Box

We are given that the ladder, five meters long, props against a wall over a box, which is 1 m on each side.

For this problem, we want to estimate the maximum height on the wall to which the ladder can reach.

 

 

Based on the picture above, there are two types of relationships depicted: similar triangles and a Pythgorean relationship.

Similar triangles yields the following: (y/1) = (1/x)

 

Pythgorean relationship:

 

We can use similar triangles and the Pythgorean relationship to estimate the solution. First, let's explore the problem using Graphing Calculator.

 

 

 

 

The graph implies that there are four possible solutions for the system of equations; however, the first quadrant is our main focus for this problem. The first quadrant is more important because we are interested in the length of x and y in first drawing depicted above, which should be positive. The solutions in the third quadrant are irrelevant because they are negative.

 

To estimate a solution, I chose to use a computer software called Graphing Calculator. I subituted (1/x) for y, expandecd the expanded the equation and multiplied the polynomial by (x^2). To determine the maximum height on the wall that the ladder reaches, it suffices to find what value makes the first polynomial zero. The second polynomial was used for this purpose. The graph of the second polynomial yield a line, and it was most important in the estimation. Since the first equation must equal zero, I tried different values of a with careful attention on how close it approached zero, but never beyond x = 0.

 

 

I have estimated that x is approximately .2586898, which implies that y is approximately 3.86563. (Y+1) is an estimate of the maximum height that the ladder touches on the wall.