Question

A ladder 10 ft long is resting against a wallIf the bottom of the ladder is sliding away from the wall at a rate of 1 ft per sec how fast is the top of the ladder moving down when the ladder is 8 ft from wall?

Answer 1

Answer:

The top of the ladder is moving 1.33 ft per second downwards

Explanations:

Let the height of the ladder be l

Let the distance of the ladder to the wall be x

x = 8 ft

Let the distance from from the top of the ladder to the bottom be d

d = 10 ft

The illustration can be shown by the diagram below:

To find the distance d, use the Pythagorean theorem:

`\begin{gathered} 10^2=l^2+x^2 \\ 10^2=l^2+8^2 \\ l^2=100-64 \\ l^2=36^{} \\ l\text{= }\sqrt[]{36} \\ l\text{ = 6 ft} \end{gathered}`

Now, to calculate the speed of the ladder from the top to the bottom, find the derivative of the equation l² + x² = 10² with respect to the time t

The equation becomes:

`\begin{gathered} 2l(dl)/(dt)+2x(dx)/(dt)=\text{ 0} \\ 2l(dl)/(dt)\text{ = -}2x(dx)/(dt) \\ (dl)/(dt)\text{ = }(-2x)/(2l)(dx)/(dt) \\ (dl)/(dt)\text{ = }(-x)/(l)(dx)/(dt) \\ \text{Note that }(dx)/(dt)=1ft\text{ per sec} \\ x\text{ = 8, l = 6} \\ (dl)/(dt)=(-8)/(6)(1) \\ (dl)/(dt)=(-4)/(3) \end{gathered}`

dl/dt = -1.33 ft/s

This means that the top of the ladder is moving 1.33 ft per second down