CALCULUS HELP: Related rates.. A painting hangs on the wall with the bottom...?

A painting, which is 3m tall, hangs on a wall with the bottom of the painting 3m above the floor. A man, whose eyes are 2m above the floor, looks at the painting while backing away at 0.5 m/sec. Find the rate (in radians/sec) at which the angle subtended by the painting at the man's eye is decreasing when the man is 3m from the wall.





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Will award 10 points for best answer (answers only with steps)CALCULUS HELP: Related rates.. A painting hangs on the wall with the bottom...?
Let x be the distance from the wall, and 胃 the (vertical) angle subtended by the painting. Then:





胃 = arctan(4/x) - arctan(1/x)


d胃/dx = 1/(1 + (4/x)虏) * (-4/x虏) - 1/(1 + (1/x)虏) * (-1/x虏)


= -4/(x虏 + 16) + 1/(x虏 + 1)





At x=3, d胃/dx = -4/25 + 1/10 = -0.06





By the chain rule, we know:


d胃/dt = d胃/dx * dx/dt





And we are given that dx/dt = 0.5, so:


d胃/dt = -0.06 * 0.5 = -0.03 radians / sec.