Question

1. persevere with problems suppose a ladder 20 feet long is placed against a vertical wall 20 feet high. how far would the top of the ladder move down the wall by pulling out the bottom of the ladder 5 feet? explain your reasoning.

Explanation:

Step1: Apply Pythagorean theorem initially

The Pythagorean theorem is \(a^{2}+b^{2}=c^{2}\), where \(c\) is the length of the ladder (hypotenuse), \(a\) is the height on the wall, and \(b\) is the distance from the wall. Initially, \(c = 20\) and \(a=20\), then \(b=\sqrt{c^{2}-a^{2}}=\sqrt{20^{2}-20^{2}} = 0\) (the ladder is just placed straight up initially). After pulling out the bottom \(b = 5\), and \(c = 20\) (length of ladder remains same).

Step2: Find the new height \(a\)

Using \(a=\sqrt{c^{2}-b^{2}}\), substitute \(c = 20\) and \(b = 5\). So \(a=\sqrt{20^{2}-5^{2}}=\sqrt{400 - 25}=\sqrt{375}=5\sqrt{15}\approx19.36\)

Step3: Calculate the distance moved down

The initial height was \(20\) feet and the new height is approximately \(19.36\) feet. The distance moved down is \(20 - 19.36=0.64\) feet.

Answer:

The top of the ladder moves down approximately \(0.64\) feet.