Question

find the speed of the top of the ladder while it is sliding down

Answer 1

The diagam of the ladder can be given as,

According to pythagoras theorem,

$x^2+y^2=l^2$

At x= 5 ft and l=19 ft. The value of y can be solved as,

$\begin{gathered} (5ft)^2+y^2=(19ft)^2 \\ y^2=361ft^2-25ft^2 \\ y=\sqrt[]{336ft^2} \\ \approx18.3\text{ ft} \end{gathered}$

Differentiate the above equation with respect to time.

$\begin{gathered} 2x\frac{dx}{dt}+2y\frac{dy}{dt}=0 \\ \text{y}\frac{dy}{dt}+x\frac{dx}{dt}=0 \end{gathered}$

Substitute the known values,

$\begin{gathered} (18.3\text{ ft)}\frac{dy}{dt}+(5\text{ ft)}(1\text{ ft/s)=0} \\ \frac{dy}{dt}(18.3\text{ ft)=-(5 ft/s)} \\ \frac{dy}{dt}=-0.273\text{ ft/s} \end{gathered}$

Therefore, the speed at which the ladder slide down is 0.273 ft/s and the negative sign indicates the downward direction of sliding.