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:

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

dl/dt = -1.33 ft/s

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