A ladder 8m long is to be moved horizontally around a cornet from a corridor 2m wide into the second corridor that right angles the first. What is the wide of the second corridor to allow the ladder go around the corner? Neglect the width of the ladder.
interesting.... suppose we assume the ladder gets stuck. |\L |_\ The ladder forms some angle with the wall. Lets call A the bottom right corner of the triangle. B the top left, C the right angle of that triangle where the hallway bends and D the point we are getting stuck on. lets call the angle at A a. The distance that B is from the opposite wall is 8 sin a the distance that the ladder goes down the hallway from D to B is 2 tan a. the width of the hallway we are trying to find then, taken up by the ladder is W = 8 sin a - 2 tan a find the a that maximizes W and we know how big the hallway needs to be. dW/da = 8 cos a - 2 sec^2 a = 0 8 cos a = 2 / cos^2 a cos^3 a = 1/4 cos a = (2)^(-2/3) a = arc cos (2)^(-2/3) ~ 0.89 radians 8 sin a - 2 tan a = 3.75 meters