Question

question: a 10 - foot ladder is leaning against a wall. if the top of the ladder slides down the wall at a rate of 2 feet per second, how fast is the bottom of the ladder moving along the ground when the bottom of the ladder is 5 feet from the wall? write your answer in the most simplified form without rounding and do not include units. provide your answer below.

Explanation:

Step1: Apply Pythagorean theorem

Let $x$ be the distance of the bottom from the wall and $y$ be the height of the top on the wall. So $x^{2}+y^{2}=10^{2}=100$. When $x = 5$, then $y=\sqrt{100 - 25}=\sqrt{75}=5\sqrt{3}$.

Step2: Differentiate with respect to time $t$

$2x\frac{dx}{dt}+2y\frac{dy}{dt}=0$. We know $\frac{dy}{dt}=- 2$ and $x = 5,y = 5\sqrt{3}$.

Step3: Solve for $\frac{dx}{dt}$

$5\times\frac{dx}{dt}+5\sqrt{3}\times(-2)=0$, so $\frac{dx}{dt}=2\sqrt{3}$.

Answer:

$2\sqrt{3}$