1 Answer

Jack Pilowsky · Answer 1 · 2023-01-28T00:39:08+0000

Answer:

$\left.\frac{\text{d}x}{\text{d}t}\right|_(h=2)\approx0.184\; \sf m/s$

Explanation:

$\boxed{\begin{minipage}{9 cm}\underline{Pythagoras Theorem} \\\\$a^2+b^2=c^2$\\\\where:\\ \phantom{ww}$\bullet$ $a$ and $b$ are the legs of the right triangle. \\ \phantom{ww}$\bullet$ $c$ is the hypotenuse (longest side) of the right triangle.\\\end{minipage}}$

Given variables:

a = length of the ladder.
h = height of the ladder's top at time t.
x = distance of the ladder from the wall at time t.

Given a = 8.9, use Pythagoras Theorem to create an equation for x² in terms of h²:

$\implies x^2+h^2=a^2$

$\implies x^2+h^2=8.9^2$

$\implies x^2+h^2=79.21$

$\implies x^2=79.21-h^2$

Differentiate with respect to h:

$\implies 2x\frac{\text{d}x}{\text{d}h}=0-2h$

$\implies \frac{\text{d}x}{\text{d}h}=-(2h)/(2x)$

$\implies \frac{\text{d}x}{\text{d}h}=-(h)/(x)$

Given:

$\frac{\text{d}h}{\text{d}t}=-0.8\; \sf m/s$

Therefore:

$\begin{aligned} \frac{\text{d}x}{\text{d}t}&=\frac{\text{d}x}{\text{d}h}* \frac{\text{d}h}{\text{d}t}}\\\\ \implies &=-(h)/(x) * -0.8\\\\ &=(0.8h)/(x) \end{aligned}$