how fast is the top of the ladder sliding down the wall when the bottom of the ladder is 6ft from the wall?

Question

A ladder 10ft long rests against a vertical wall. If the bottom of the ladder slides away from the wall at a rate of 1ft/s how fast is the top of the ladder sliding down the wall when the bottom of the ladder is 6ft from the wall?

Answer 1

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

     y = distance from base of wall to top of ladder at time            t

At any time t, we have a right triangle with legs x and y and with hypotenuse 10.

 

By the Pythagorean Theorem, x² + y² = 100  (*)

When x = 6, 36 + y² = 100.  So, y² = 64.  

Therefore, y = 8.

 

Differentiate (*) implicitly with respect to t:

 

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

2(6)(1) + 2(8)(dy/dt) = 0

16(dy/dt) = -12

dy/dt = -0.75

 

When x = 6 ft, the ladder is sliding down the wall at the rate of 0.75 ft/sec.

Answer 2

Hi Magi,

 

       Let x= distance from the wall

             dx/dt rate of change of x =1ft/sec

             y= distance from the floor to the top of the ladder=(100-x²)^1/2

              dy/dt= speed of the top of the ladder=-(x/(100-x²)^1/2)dx/dt=-.75ft/sec. at x=6 and dx/xt=1 ft sec

               Used the chain rule for differentiation of f(g(x)).

     Hope this helps

 

Jim