40/3 feet /second fast the top of the ladder sliding down the wall.
What is ladder?
A ladder is a piece of gear used for climbing up or down objects that consists of repeated bars or steps (rungs) between two upright lengths of metal, wood, or rope.
Typically, these ladders work best for outdoor tasks like roof access or exterior painting. A straight ladder's reach is typically two feet higher than its height.
Given:
A 10 foot ladder is leaning against a wall when the base begins to slide away from the wall. when the base is 8 feet from the wall, the base is moving at the rate of 4 feet per second.
We have dx/dt = 4 feet per second
We want to find dy/dt.
From given information,
x^2 + y^2 = 10^2 ..(1)
When x = 8
⇒ y = √(10^2 - 8^2)
y = √(100 - 64)
y = √36
y = 6 feet
Now differentiate equation (1) with respect to t
[tex]2x\frac{dx}{dt} +2y\frac{dy}{dt} = 0\\ x\frac{dx}{dt} + y\frac{dy}{dt} = 0\\ \frac{dy}{dt} = -\frac{x}{y} \frac{dx}{dt}[/tex]
Plug x = 8, y = 6 and dx/dt = 10
⇒
[tex]\frac{dy}{dt} = -\frac{8}{6}(10)\\ \frac{dy}{dt} = -\frac{40}{3}[/tex]
Hence, 40/3 feet /second fast the top of the ladder sliding down the wall.
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