A person is carrying a ladder horizontally and intends to carry the ladder around the L-shaped corner where a hallway of width 2 feet and a hallway of width 4 feet meet (see applet below). Assuming the ladder is a theoretical ladder of negligible thickness, what is the longest ladder that may be carried around the corner?

(a) Letting x represent the distance from the corner to the point of contact of the right end of the ladder with the hallway wall (see applet), write a function giving the length of the ladder in terms of x. Hint: To set up this function, you will need to use the proportionality of ratios derived from similar triangles. The choice of that proportional relationship will make the derivative equation you must solve very easy to solve or very hard to solve. Make sure one of the triangles you use is the "big" one, i.e. the triangle the ladder makes with the corner.

      $\displaystyle {L}{\left({x}\right)}=$ Preview Question 6 Part 1 of 3  

(b) The maximum allowable length for the ladder is feet. Enter an exact answer. Preview Question 6 Part 2 of 3  

(c) Enter an approximation for the value from part (b).

      feet