CASE STUDY CHALLENGE

AOD: Ladder Optimization Problem

A ladder of fixed length ‘h’ is to be placed along a wall such that it is free to move along the height of the wall. The ladder, the wall, and the ground form a right-angled triangle.

Ladder against wall diagram

Variables defined:

  • • h = Fixed length of the ladder (Hypotenuse)
  • • x = Height on the wall
  • • y = Distance of the foot of the ladder from the wall

Answer the following:

(i) Express the distance (y) in terms of ‘h’ and ‘x’. Also, write an expression for the Area (A) of the right triangle in terms of ‘h’ and ‘x’. [1]
(ii) Find the derivative of the area (A) with respect to x, and find its critical point. [1]
(iii) (a) Show that the area (A) is maximum at the critical point.

— OR —

(b) If a ladder of length 5 m is being pulled towards the wall such that the rate of decrease of distance (y) is 2 m/s, at what rate is the height (x) increasing when the foot is 3 m away from the wall?

 [2]
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