math problem????????

Question

i dont get Related rate problems at all
can someone just teach me the steps of solving any type of related rate problem
i know derivatives but not related rates

Answer 1

One of the most common type of related rate problems involves a ladder leaning against a wall:
Suppose that there is a 10-meter ladder leaning against the wall of a building, and the base of the ladder is sliding away from the building at a rate of 3 meters per second. How fast is the top of the ladder sliding down the wall when the base of the ladder is 6 meters from the wall?

Calling the distance of the base of the ladder from the wall x and the height of the ladder on the wall y, the ladder, the wall, and the ground represent the sides of a right triangle with side lengths x, y, and 10 (the hypotenuse). The object is to find the rate of change of y with respect to time when x = 6. It is given that when x = 6, the rate of change of x is 3 meters per second. This rate of change is positive because the distance x is increasing.

An equation relating the three sides of a right triangle is the Pythagorean Theorem, a^2+b^2 = c^2. In this case, the equation that relates x and y is x^2+y^2 = 10^2. Differentiating both sides of this equation with respect to time (t) yields
(d/dt)(x^2 + y^2) = (d/dt)(100)

which when solved for the wanted rate of change, dy/dt, gives us
dy/dt = -x(dx/dt)/y

It is given that when x = 6, dx/dt = 3. Due to the pythagorean theorem, y = 8. Plugging these values into the equation gives us the answer:
dy/dt = -6(3)/8 = -9/4

The top of the ladder is sliding down the wall at a rate of 9⁄4 meters per second.