A 13-foot ladder is leaning against a house when its base starts to slide away. by the time the base is 12ft ?
A 13-foot ladder is leaning against a house when its base starts to slide away. by the time the base is 12ft from the house, the base is moving at the rate of 5ft/sec
a. how fast is the top of the ladder sliding down the wall at that moment?
b. At what rate is the area of the triangle formed by the ladder, wall, and ground changing at the moment?
c. at what rate is the angle between the ladder and the ground changing at that moment?
By the Pythagorean theorem, h = √(169 - b²)
dh/db = -b / √(169 - b²)
Now use the chain rule: dh/dt = db/dt dh/db ........... db/dt is given as 5
dh/dt = 5 (-12) / √(169 - (12)²)
dh/dt = -12
Answer (a): -12 ft/sec (minus indicating that h is decreasing)
A = 1/2 b h
dA/db = (b/2) dh/db + h/2
= (b/2) (-b / √(169 - b²)) + √(169 - b²)/2
Again, use the chain rule: dA/dt = dA/db db/dt
dA/dt = 5 [ (12/2) (-12 / √(169 - (12)²)) + √(169 - (12)²)/2 ]
= -119/2 ft²/s
Answer (b): -119/2 ft²/s (minus indicating that A is decreasing)
cosθ = b/13
-sinθ dθ/db = 1/13
dθ/db = -1/(13 sinθ) = -1/(√(169 - b²))
Again, use the chain rule: dθ/dt = dθ/db db/dt
dθ/dt = 5 [ -1/(√(169 - (12)²)) ]
= -1
Answer (c): -1 radian/sec (minus indicating that θ is decreasing)
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