What is the related rate of the problem

Question

A ladder 13−−√13 meters long rests on horizontal ground and leans against a vertical wall. The foot of the ladder is pulled away from the wall at the rate of 0.30.3 m/sec. How fast is the top sliding down the wall when the foot of the ladder is 22 m from the wall? (Do not need "-" in the answer.)

Answer 1

In related rates problems, often the first step is to determine which things are changing (the rates that are related), and which things are constant. In this example, we know that size of the ladder (c) isn't changing. However, as the ladder slides, the foot of the ladder's distance from the wall (a) is increasing as the height of the ladder against the wall (b) is decreasing. What equations do we have in our tool belt that use the 3 sides of a triangle?

The Pythagorean Theorem! If we start with a² + b² = c², we can find the starting height of the ladder against the wall, since we know the size of the ladder (c) and that it started 22m from the wall (b). When we take the derivative of both sides with respect to time, we're left with:

2a(da/dt) + 2b(db/dt) = 0

c² becomes 0 because after all, the ladder's length is a constant! With this setup, you can now plug in the initial values from your problem set up and use algebra to isolate the one remaining rate.

The key to solving this problem was recognizing what pieces you were given and then finding a geometric relationship that allowed you to get the rates you would need.

Answer 2

if the ladder is 13sqr13 meters long (if that's what you meant)

then when it is 22 feet from the wall's base

the top of the ladder is about 41.39 m high= y

constrct a right triangle

use the pythagorean theorem

y = sqr((13sqr13)^2 -22^2) = about 41.39

h^2 = x^2 + y^2

take derivatives with respect to time, h'=0, x'=.303, x=22

2hh' = 2xx' + 2yy'

divide by 2, plug in known values

0 = 22(.303) + 41.39y'

solve for y'= how fast the top of the ladder is sliding down the wall

y'=about -0.161 meters per second

<0 indicating it's going down