We have a weekly assignment and the teacher posts solution but doesn't EXPLAIN how she got the answer. just gives you the answer.
So I got this question wrong and I need help on how the answer was found..
The length of a rectangle is increasing at 2 m/s and with is increasing 1 m/s. when the length is 5m and width is 3 m how fast is the area increasing?
a ladder 10 meters long is leaning against a wall, with the foot of the ladder 8 meters away from the wall. if the foot of the ladder is being pulled away from the wall at 3 meters per second, how fast is the top of the ladder sliding down the wall?
Best Answer
I'll solve your second problem. Which is better stated as:
A ladder 10 meters long is sliding against a wall. If the foot of the ladder is being pulled away from the wall at 3 meters per second, how fast is the top of the ladder sliding down the wall when the foot of the ladder is 8 meters away from the wall. ?
Identify the variables in the problem. What is changing? It is very important to introduce and name the variables here.
The height $h$ from the top of the ladder to the floor and the length $l$ from the bottom of the ladder to the wall are changing.
Ask yourself: "What rates of change do I know?" and "What rate of change is being asked for?".
You know $l$ is increasing at a rate of 3, so ${dl\over dt}=3$.
You need to find the rate of change of $h$ when $l=8$. So, you want to find ${dh\over dt } \Bigl |_{l=8}$.
Ok, we need to find a rate of change of $h$ and we have these variables $l$ and $h$...
By the Pythagorean Theorem: $$ \tag{2}l^2+h^2=100. $$
But, we want to find $h'$. How to get that?
You are given ${dl\over dt}=3$ and $l=8$ and you can calculate $h=\sqrt{100-64}=6$.
${dh\over dt}\Bigl |_{l=8}$ is what we are trying to find.
Now substitute this information into (3): $$ 2\cdot8\cdot 3+2\cdot6\cdot{dh\over dt}\Biggl |_{l=8} =0, $$ and solve for ${dh\over dt}\Bigl |_{l=8}$: $$ {dh\over dt} \Biggl|_{l=8}=-{ 48\over 12}=-4. $$
I should have used units throughtout, but was to lazy to... Note that the answer should be negative since the top of the ladder would be moving down.