一道数学题,关于微分的.One of NASA's old satelites is falling back to the earth.When it lands,it will spash down into the ocean,making a circular ripple that will travel outward at a speed of 0.55m per second.Find the rate at which the area

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一道数学题,关于微分的.
One of NASA's old satelites is falling back to the earth.When it lands,it will spash down into the ocean,making a circular ripple that will travel outward at a speed of 0.55m per second.Find the rate at which the area within the circle of the first ripple is increasing after 85 seconds.

不考虑地球表面曲率影响,将洋面波当作平面波来计算.首先分析题目已知条件知圆半径r与时间t的关系:dr/dt=0.55,初始条件:当t=0时r=0.所有单位均取SI单位制.
解微分方程得到:r=0.55t
而面积S=πr²,两边取微分得到:
dS=2πrdr
将已知微分方程代入上述微分式得到:
dS=2πr(0.55dt)=1.1πrdt
整理得到:dS/dt=1.1πr
再把r与t的函数关系代入得到:
dS/dt=0.605πt
t=85s时的面积增长率为:
0.605π×85=161.5564 m²/s