APPLICATION OF DERIVATIVES - RELATED RATES

Question

A 14-foot ladder is leaning against a wall. If the top of the ladder slips down the wall at a rate of 2 ft/s., how fast will the end be moving away from the wall when the top is 6 ft. above the ground.

Answer 1

distance of the bottom of the ladder to the wall, when the top is 6 feet from the bottom of the wall is:

B = square root of 14^2 - 6^2 = sqr(196-36) =sqr160 = 4sqr10 = 12.65 about feet

Use the Pythagorean Theorem

H^2 = B^2 + T^2 where T=6, B=12.65 and H = 14 feet

take the derivative with respect to time

2HH' + 2BB' + 2TT'

divide by 2

HH' + BB' + TT' where H' =0, T'=-2, H=14, B=12.65, T=6

H(0) = 0 = B' + 6(-2)

solve for B'

12.65 B' = 12

B' = 12/12.65 = about 0.95 feet per second = how fast the end of the ladder is moving away from the wall

B'= about 1 foot per second

this type of problem is in calculus textbooks under the section on derivatives, with usually a couple pages on the topic related rates."