Question

Use precise mathematical language to justify and explain each mathematical process. A ladder is placed against a wall to reach the highest window of a house. A right angle is formed

by the wall of the house and the ground.
1. Sketch a diagram to represent the situation.
2. If the ladder is 20 feet long and the window is 12 feet high, how far is the base of the ladder
from the wall?
3. Could a 15 foot ladder be placed at the same 12 feet high window, but 10 feet from the house?
Justify your answer.

Answer 1

Using the Pythagorean theorem, we found the distance from the base of a 20-foot ladder to the wall is 16 feet. A 15-foot ladder cannot reach a 12-foot high window if placed 10 feet away from the house, as the maximum height it could reach is only approximately 11.18 feet.

Step-by-step explanation:

Resolving a Right Triangle in a Practical Scenario

When a ladder is placed against a wall, forming a right angle with the ground, we can depict this as a right triangle in a sketch. Given that the ladder is 20 feet long and reaches a window 12 feet high, we can solve for the distance from the base of the ladder to the wall using the Pythagorean theorem: √(20² - 12²) feet. When calculated, this distance is 16 feet.

As for a 15-foot ladder, it is impossible for it to reach the same 12-foot-high window if it is 10 feet away from the house. Using the Pythagorean theorem again, the maximum height a 15-foot ladder could reach at that distance is only √(15² - 10²) feet, which is 11.18 feet, not sufficient to reach the 12-foot high window.