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)

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