Associated Topics || Dr. Math Home || Search Dr. Math

```
Date: 11/30/2001 at 11:44:30
From: Michelle Rittenhouse

Two hallways meet to form a right angle. One is 8 ft. wide and the
other is 4 ft. wide. What is the longest ladder that can go around
the corner where the hallways meet? The ladder is carried
horizontally. Disregard the width of the ladder.
```

```
Date: 11/30/2001 at 12:37:41
From: Doctor Rob

Thanks for writing to Ask Dr. Math, Michelle.

I drew this diagram:

A       x+4      C
---o---------------o
:\              |
: \             |
:  \            |
8:   \           |
:    \          |
:     \         |
:   x  \        |
---o-------o       |8+32/x
E      D|\      |
| \     |
|  \    |
32/x|   \   |
|    \  |
|     \ |
|      \|
F o.......o B
|   4   |

Let the length of DE be x. Using similar triangles ADE and DBF, you
will find that DF = 8*4/x = 32/x. Then using the Pythagorean theorem
on triangle ABC, the length L of AB is given in terms of x by

L = sqrt([x+4]^2+[8+32/x]^2),
= sqrt([x+4]^2*[1+(8/x)^2]),
= sqrt(x^2+64)*(x+4)/x.

To find the minimum value of L as the length x changes, which is the
maximum length the ladder can have, you can differentiate this with
respect to x, set that equal to zero, and solve for x. The equation I
got after simplifying was

(x^3-256)/(x^2*sqrt[x^2+64]) = 0.

Solve this for x, and then you can compute the corresponding value of
L. My answer came out between 15 and 20 feet.

Feel free to write again if I can help further.

- Doctor Rob, The Math Forum
http://mathforum.org/dr.math/
```
Associated Topics:
High School Calculus
High School Geometry
High School Practical Geometry
High School Triangles and Other Polygons

Search the Dr. Math Library:

 Find items containing (put spaces between keywords):   Click only once for faster results: [ Choose "whole words" when searching for a word like age.] all keywords, in any order at least one, that exact phrase parts of words whole words

Submit your own question to Dr. Math
Math Forum Home || Math Library || Quick Reference || Math Forum Search