SOLUTION: A 17 foot ladder is leaning against a wall. If the top slips down the wall at a rate of 4 ft/s, how fast will the foot be moving away from the wall when the top is 14 feet above th

Algebra ->  Finance -> SOLUTION: A 17 foot ladder is leaning against a wall. If the top slips down the wall at a rate of 4 ft/s, how fast will the foot be moving away from the wall when the top is 14 feet above th      Log On


   

Question 1160398: A 17 foot ladder is leaning against a wall. If the top slips down the wall at a rate of 4 ft/s, how fast will the foot be moving away from the wall when the top is 14 feet above the ground?
Answer by greenestamps(13200) About Me  (Show Source):
You can put this solution on YOUR website!


x is the distance the foot of the 17-foot ladder is from the wall when the height the ladder reaches on the wall is y:

x+=+sqrt%2817%5E2-y%5E2%29+=+%28289-y%5E2%29%5E%281%2F2%29

Differentiate to find the rate of change in x for the given rate of change in y.

Square the given equation to avoid differentiating the square root function.

x%5E2+=+289-y%5E2
2x%2A%28dx%2Fdt%29+=+-2y%2A%28dy%2Fdt%29

dy/dt is given as -4 (ft/sec); when y=14, x is sqrt%28289-196%29+=+sqrt%2893%29.

2%2Asqrt%2893%29%28dx%2Fdt%29+=+-2%2814%29%28-4%29
dx%2Fdt+=+112%2F%282%2Asqrt%2893%29%29+=+56%2Fsqrt%2893%29

When the top of the ladder is 14 feet above the ground, the foot is moving away from the wall at a rate of 56/sqrt(93) ft/sec, or about 5.8 ft/sec.