calculus integration question 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?
 A: Let's look at your first question: let $w(t)$ be the width, $l(t)$ the height, and $A(t)$ the area of your rectangle, as a function of time.
We know from geometry that $A(t) = l(t)w(t)$.
We want to find the rate at which the area is changing: in other words, the derivative of area with respect to time. Using the product rule,
$$A'(t) = l'(t)w(t)+l(t)w'(t).$$
To evaluate the formula on the right, we need to know (from left to right): the rate of change of the length, the current width, the current length, and the rate of change of the width. These were given in the problem, so
$$A'(t) = 2\ \textrm{m/s} \cdot 3\ \textrm{m} + 5\ \textrm{m} \cdot 1\ \textrm{m/s} = 11\ \textrm{m}^2\textrm{/s}.$$
(Sanity check: the units of the answer, meters squared per second, make sense for measuring the rate of change of area.)
Can you now do the ladder problem on your own?
A: 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 $h$...

*Write an equation relating the variables.By the Pythagorean Theorem:
$$
\tag{2}l^2+h^2=100.
$$
But, we want to find $h'$. How to get that?

*Implicitly differentiate (2) with respect to time $t$ to obtain: 
$$
\tag{3}2l{dl\over dt}+2h{dh\over dt} =0.
$$

*Now substitute what you know into (3) and solve for what you don't: 

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.
