Rates of change

Question

A ladder AB of length 5m has one end A leaning against a vertical wall. The other end B rests on the horizontal ground. When A is at a height of  4m , it slides down the wall at the rate of 2m/s. How fast is the other end, B, sliding along the horizontal ground?

Answer 1

Let x = distance from base of the ladder to the wall at time t

      y = height of ladder above the ground at time t

 

Given:  dy/dt = -2

Find:  dx/dt when y = 4

 

Solution:  The triangle formed by the ladder, wall, and ground is a                         right triangle.

 

                  By the Pythagorean Theorem, x² + y² = 25

 

                  Differentiate implicitly with respect to t to get

 

                        2x(dx/dt) + 2y(dy/dt) = 0

 

                        dx/dt = -y(dy/dt)/x  = -4(-2)/x  = 8/x

 

                            Since x² + y² = 25 and y = 4, x = 3

 

                        So, dx/dt = 8/x = 8/3 meters/sec