微积分高手来做做Aspartofanemergencyoperationtocontainanuclearspill,alargecone(withashapeasshowninthefigure)ofheight100mandradius50missuspendedfromitstoppointandloweredintothewaterattherateof0.1m/s.(As

Question

微积分高手来做做
As part of an emergency operation to contain a nuclear spill,a large cone (with a shape as shown in the figure) of height 100m and radius 50m is suspended from its top point and lowered into the water at the rate of 0.1m/s.(As it is lowered,air escapes at the top of the cone and water enters at the bottom so that the water levels inside and outside the cone are the same.)
(a) What is the radius of the cone at the water‟s surface when the top of the cone is at a height h above the water surface?
(b) At what rate is the enclosed air volume in the cone changing when the top of the cone is at a height of 30m above the water level?
Note:To summarise the Formulae and Tables booklet (page 6),a “right circular cone” with base radius r,height h,and „sloping height‟ l has a curved surface area = pi r l and a volume = 1/3 pi r^2 h.
大意是一个圆锥体高h=100 地步的 半径是 r=50 在水面上头朝上往下沉 下面没低 所以圆锥体里的水会和旁边的水面一个水平.下沉的速度是 0.1m/s (dh/dt=0.1m/s) 给了圆锥体的面积和体积的公式 （a）当圆锥体的顶部和水面是h 要求半径r是什么?
这题用体积的公式把 r=(3V/πh)^1/2 就可以了吗?
(b)是要求体积的速率（dV/dt) 当h=30m 在水面上
dv/dt=dh/dt x dV/dh 但是dV/dh=1/3πr^2 h就没了 没法带30进去?
可以面积里的l（ 圆锥体的斜长度） r^2=l^2-h^2 放进 dV/dh 的r^2里面 但是又不知道l是多少.

Answer 1

1.三角形相似,h/100=r/50
2.r由1用h表示,就可以了,直接微分.