微积分应用问题
问题描述：
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.
答
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^3/27h=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...