Expert: Abe Mantell - 5/29/2007

Question
1- A ladder 25 feet long is leaning against the wall of aa house. The base of the ladder is pulle away from the wall at a rate of 2 feet per second. How fast is the top moving down the wall when the base of the ladder is (a) 7 feet, (b) 15 feet, and (c) 24 feet from the wall?

2- Consider the right formed by the moving ladder, the side of the house, and the ground in Exercise 1. When  the base is 7 feet from the wall, find the rate at which the ares of the triangle is changing

Answer
1. Let x=the distance the base of the ladder is from the bldg, and y=the height.  Thus, x^2 + y^2 = 25^2 = 625
differentiating wrt 't': 2x x' + 2y y'=0 ==> y'=-(x/y) x'
y=sqrt(625-x^2)...y'=-(x/sqrt(625-x^2)) x'
(a) x=7 ==> x'=-(7/24) 2 ft/sec = -7/12 ft/sec
(b) x=15 ==> x'=-(15/20) 2 ft/sec = -3/2 ft/sec
(c) x=24 ==> x'=-(24/7) 2 ft/sec = -48/7 ft/sec

2. Area, A=xy/2...differentiating wrt 't' gives
A'=xy'/2 + x'y/2 when x=7, y=24, x'=2, y'=-7/12
so A'=7(-7/12)/2 + 2(24)/2 = 527/24 sq. ft./sec