Answer to Question #250811 in Calculus for Sam

Question

9. A ladder 20 ft long is leaning against an embankment inclined 60 ° to the horizontal. If the bottom of the ladder is moved horizontally toward the embankment at 1 ft/sec, how fast is the top of the ladder moving when the bottom is 4 ft from the embankment?

Answer 1

Consider the diagram below

Let AC be x ft and BC be y ft

$x^2+y^2=20^2\\ y^2=20^2-x^2\\ 2y \frac{dy}{dt}=-2x\frac{dx}{dt}\\ Also \\ \frac{dx}{dt}=1 \implies 2y \frac{dy}{dt}=-2x*1 \implies \frac{dy}{dt}= -\frac{x}{y}\\ y^2=400-16 \implies y =\sqrt{384}\\ \frac{dy}{dt}= -\frac{4}{\sqrt{384}}=-0.2041$

Therefore the top of the ladder is moving downwards at 0.2041 ft/s