Question

Upper A 20​-foot ladder is leaning against a building. If the bottom of the ladder is sliding along the pavement directly away from the building at 2 ​feet/second, how fast is the top of the ladder moving down when the foot of the ladder is 2 feet from the​ wall?

Answer 1

Answer:

  -(2√11)/33 ≈ -0.201 ft/s

Step-by-step explanation:

You want the rate at which a 20 ft ladder is sliding down a wall when its base is 2 ft from the wall and moving away at 2 ft/s.

Height

The height of the ladder on the wall is given by the Pythagorean theorem. If x is the distance from the wall, and y is the height, we have ...

  x² +y² = 20²

  y = √(20² -x²)

Rate of change

The derivative with respect to time is ...

  $y'=\dfrac{-2x\cdot x'}{2\sqrt{400-x^2}}=\dfrac{-x\cdot x'}{\sqrt{400-x^2}}$

When x' = 2 and x = 2, this becomes ...

  $y'=\dfrac{-2(2)}{\sqrt{400-2^2}}=\dfrac{-2}{3\sqrt{11}}=-\dfrac{2\sqrt{11}}{33}\approx-0.201$

The ladder is moving down the wall at -(2√11)/33 ≈ -0.201 ft/s.