Answer
$ - \frac{15\times 5}{8}=-9.375$ ft/s
Work Step by Step
Reference to the right-angled triangle ABC in fig Ex 16
Length of the ladder = AB=z=17 ft
Height of the wall =AC=y=8 ft
Distance from the foot of the ladder to the foot of the wall=BC=x ft
By Pythagorean theorem in the right-angled triangle ABC
$AC^2+BC^2=AB^2$ Or
$y^2+x^2=z^2$ ............... eq (1)
Putting $y=8$ft, $z=17$ft in equation (1)
$8^2+x^2=17^2$
$x^2=17^2-8^2=225$ Or
$x=15$ft
Taking derivative with respect to t of equation (1)
$\frac{d(y^2)}{dt}+\frac{d(x^2)}{dt}=\frac{d(z^2)}{dt}$ ....................... eq (2)
Since z being length of ladder is constant $\Longrightarrow$ , $\frac{d(z^2)}{dt}=0$
Equation (2) may be written as
$\frac{d(y^2)}{dy}\frac{dy}{dt}+\frac{d(x^2)}{dx}\frac{dx}{dt}=0$
$2y\frac{dy}{dt}+2x\frac{dx}{dt}=0$
$y\frac{dy}{dt}=-x\frac{dx}{dt}$ ..................... eq (3)
Putting $y=8$ft, $x=15$ft, $\frac{dx}{dt}=5$ ft/s
$8\frac{dy}{dt}=-15 \times 5$
$\frac{dy}{dt}= -\frac{15\times 5}{8}=-9.375$
