微积分应用问题Water is leaking out of an inverted conical tank at a

来源：学生作业帮编辑：作业帮分类：英语作业时间：2024/11/06 08:31:12

微积分应用问题
Water is leaking out of an inverted conical tank at a rate of 6500 cm3/min at the same time that water is being pumped into the tank at a constant rate.The tank has height 6 m and the diameter at the top is 4 m.If the water level is rising at a rate of 20 cm/min when the height of the water is 2 m,find the rate at which water is being pumped into the tank.

圆锥体半径与高的比为4/2/6=1/3
设水高h,则水面半径R=h/3
圆锥体内水的体积V=лR^2h/3=лh^3/27
h=3л^(-1/3)V^(1/3),
h'(t)=л^(-1/3)V^(-2/3)V'(t),
V'(t)= h'(t)л^(1/3)V^(2/3),
V'(t0)=0.2л^(1/3)(л2^3/27)^(2/3)=0.8л/9,
所求注入水流速
v=V'(t0)+6500e-6 (m^3/min)
=0.8л/9+0.0065 (m^3/min)
=0.28575268(m^3/min)
=285752.68(cm^3/min)

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