Problem 235: Ladder logic

Could be a long climb to this problem’s solution. Principal T. Ruletheroost, a stickler for neat appearances, ordered Doc, the high school janitor, to replace several busted lights in the gymnasium.

“But Mr. R.,” replied Doc, “I’ll need the 40-foot ladder from the tool shed.”

“So, what’s the problem?” asked the principal, waiting for Doc to launch into one of his mathematical dissertations.

Doc, who was an associate professor at Hawkeye U. until caught pilfering paper clips, began his oration, “The ladder (of fixed length) can’t make the turn from the 16-foot-wide hallway into the 9-foot-wide corridor leading to the gym. Now, I could cut some off,” he speculated, “but not too much, or it won’t rest against the overhead joist.”

How much of the wooden ladder must be removed so Doc can carry it by one rail (level with the floor) around the corner and it will still lean against the 30-foot overhead superstructure at a safe angle of 68° with a minimum overhang of two feet? (Neglect the width of the ladder and assume the 9-foot corridor is long enough that the ladder will straighten out completely before it reaches the gymnasium doors.)

Hint: Find the minimum length of the ladder first – a change in variables will ease the calculations.

Solution to the March problem, 234: Hat trick