Question

A 15 foot ladder is leaning against one wall of an alley 9 ft wide. The ladder slips, its top sliding down the wall, it’s foot sliding across the alley and striking the opposite wall at a speed of 4 ft/sec. how fast is the top of the ladder falling at that instance? (Related rates problem)

Answer 1

This is question of differential equation
The ladder will form a right triangle with the wall and floor
let base of the triangle be x and perpendicular be y
so x² + y² = 15², by Pythagoras
now y=12 when x=9 (given) ,
differentiating x² + y² = 15²
2x dx/dt +2y dy/dt = 0
x dx/dt + y dy/dt = 0
given dx/dt= 4 ft/sec
9*4 + 12 dy/dt = 0
dy/dt = - 3 ft/sec