Ladder
Calculus
Level
2
A ladder is 20 feet long and leaning against a vertical wall. The top of the ladder is pulled vertically at 2fps. How fast the bottom of the ladder is sliding when the top is 10 ft away from the horizontal ground?
-1.15 fps
-3.15 fps
-2.15 fps
-4.15 fps
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1 solution
If the ladder has length of 20 ft, then its bottom's distance, x, from the wall at the moment the top's distance is y ft. from the floor is expressible as:
x = ( 2 0 2 − y 2 ) = ( 4 0 0 − y 2 ) (i).
The rate at which the bottom is moving respective to the top is just the related-rates problem:
d x / d t = ( d x / d y ) ( d y / d t ) (ii)
where d y / d t = 2 f t / s e c . Solving for (ii) at the moment y = 10 ft. gives:
d x / d t = [ − 2 y / 2 ∗ 4 0 0 − y 2 ] ∗ 2 = − ( 2 ∗ 1 0 ) / 4 0 0 − 1 0 2 = − 2 / 3 = − 1 . 1 5 f t / s e c
or choice C.