A15 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

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