# How fast is the top of the ladder sliding down the wall?

A 20ft ladder is leaning against a wall and begins to slide. How fast is the top of the ladder sliding down the wall as the instant of time when the bottom of the ladder is 12ft from the wall and is sliding away from the wall at a rate of 5ft/sec.

Relevance

Use the Pythagorean theorem to write the position of the top of the ladder as a function of the position of the bottom of the ladder.

Take the derivative of this function with respect to time.

Your result will include x and dx/dt, which were given in the description.

• The ladder forms a right angles triangle with the wall.

let b = base of ladder from wall

and h = height ladder from ground

we have, b² + h² = 20² ......(1)

h² = 400 - b²

h = √(400-b²)

Derive (1) with respect to time

2b db/dt + 2h dh/dt = 0

2b db/dt + 2√(400-b²) dh/dt = 0

b db/dt + √(400-b²) dh/dt = 0

dh/dt = -(b db/dt) /√(400-b²)

dh/dt = -(12*5) /√(400-12²)

dh/dt =-3.75 ft/s    ← rate at which top of the ladder is sliding down the wall