Problem solving

Question

A wall is 15 ft high and 10ft from a house. Find the length of the shortest ladder which will just touch the top of the wall and reach a 20.5 ft window above the ground. I used the Law of Cosine formula and got 6.63. Did I complete the problem correctly or should I have solved it another way?

Answer 1

It sounds like you're in a situation like this:

 

 

        /|

 y   /   |

    /     |  20.5

  /|      |

/  | 15 |

-------------------

  x     10

 

 

You want to lie a ladder down across the top of a wall so that it goes from ground, to wall, to house window. The wall is 15 feet high, the window is 20.5 feet up, and the wall is 10 feet from the house.

 

In that case, the ladder definitely can't be less than 7 feet long... It wouldn't even reach from the wall to the window! Also, the law of cosines should be used for non-right triangles where you know either three side lengths or two lengths and an angle, and you actually have two right triangles here. So, let's figure out another way to solve this problem.

 

The situation actually involves two triangles, one defined by the ground from ladder bottom to house, the ladder, and the house wall; and the other defined by the ground from ladder bottom to wall, the part of the ladder from bottom to wall, and the wall itself. The triangles are similar because they share one angle (the one where the bottom of the ladder meets the ground) and have the same measure for another (the right angles where wall and house wall meet the ground), meaning that the third angles must equal each other as well. So, by AAA similarity, the triangles are similar.

 

That means we can set up proportions between corresponding sides, where x = ground from ladder bottom to wall, and cross multiply to solve for x:

 

(wall height) / (window height) = (ground from ladder bottom to wall) / (ground from ladder bottom to house)

15/20.5 = x/(10+x)

150 + 15x = 20.5x

150 = 5.5x

27.27 = x

 

That means the length of the ground from ladder bottom to wall is 27.27 feet, and the length of the ground from ladder bottom to house is 10+27.27 = 37.27 feet.

 

You can solve for the length of the ladder, y, using the Pythagorean Theorem on the larger triangle:

 

(ladder length)² = (ground from ladder bottom to house)² + (window height)²

y² = 37.27² + 20.5²

y² = 1389.05 + 420.25

y² = 1816.77

y = 42.62

 

So, the ladder's length is approximately 42.62 feet.