Johnson Gibson
05/08/2024 · Primary School
27. Moving Ladder A ladder 25 feet long is leaning against the wall of a house (see figure). The base of the ladder is pulled away from the wall at a rate of 2 feet per second. (a) How fast is the top of the ladder moving down the wall when its base is 7 feet, 15 feet, and 24 feet from the wall? (b) Consider the triangle formed by the side of the house, the ladder, and the ground. Find the rate at which the area of the triangle is changing when the base of the ladder is 7 feet from the wall.
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(a) 当 \( x = 7 \) 英尺时,梯子顶部下移速度为 \( -\frac{7}{12} \) 英尺/秒。
(b) 当 \( x = 15 \) 英尺时,梯子顶部下移速度为 \( -\frac{3}{2} \) 英尺/秒。
(c) 当 \( x = 24 \) 英尺时,梯子顶部下移速度为 \( -\frac{48}{7} \) 英尺/秒。
当梯子底部离墙 7 英尺时,三角形面积变化率为 \( \frac{1}{2} \left( 7 \frac{dy}{dt} + y \frac{dx}{dt} \right) \)。
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