Please answer the following. Sorry i am very low on points right now. There are

Question

Please answer the following. Sorry i am very low on points right now. There are two seperate questions here

The top of a 26 foot ladder, leaning against a vertical wall, is slipping down the wall at the rate of 5 feet per second. How fast is the bottom of the ladder sliding along the ground away from the wall when the bottom of the ladder is 10 feet away from the base of the wall? A boat is pulled into a dock by a rope attached to the bow (front end) of the boat and passing through a pulley on the dock that is 4 m higher than the bow of the boat. If the rope is pulled in at a rate of 3 m/s, at what speed is the boat approaching the dock when it is 10 m from the dock?

Explanation / Answer

(a)

when the ladder is 10 feet from the wall, it is sqrt(26^2-100) feet up the wall.

The ladder is 24 feet up the wall.

d^2 = x^2 + y^2

2d * dd/dt = 2x dx/dt + 2y dy/dt

dd/dt = 0 because the ladder isn't changing length.

2x dx/dt = -2y dy/dt

2(10) dx/dt = -2(24)(-5)

dx/dy = 12 ft/s

(b)

d = length of rope

x = distance from dock

y = height of pulley

d^2 = x^2 + y^2

d^2 = (10)^2 + (4)^2

d = sqrt(16)

2d dd/dt = 2x dx/dt + 2y dy/dt

dy/dt = 0 because the pulley is at a constant height.

2(sqrt116) (-3m/s) = 2(10) dx/dt

dx/dt = (-3/5)sqrt(29) m/s

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