作业帮关于微积分的问题（Related rates）
来源：学生作业帮 编辑：作业帮 分类：英语作业 时间：2024/05/05 20:24:41
关于微积分的问题（Related rates）
If a tree trunk adds 1/4 of an inch to its diameter and 1 foot to its height each year,how rapidly is its volume changing when its diameter is 3 feet and its height is 50 feet (assume that the tree trunk is a circular cylinder).
这个问题或者这类Related rates问题是不是只能用微积分的方式解决而不能用algebra?具体是用calculus怎么解决的?为什么只能用calculus?还有可不可以介绍一下关于Related rates的知识?我是要做英文的presentation.所以最好是英文.不过中文也可以,不需要太长.
If a tree trunk adds 1/4 of an inch to its diameter and 1 foot to its height each year,how rapidly is its volume changing when its diameter is 3 feet and its height is 50 feet (assume that the tree trunk is a circular cylinder).
这个问题或者这类Related rates问题是不是只能用微积分的方式解决而不能用algebra?具体是用calculus怎么解决的?为什么只能用calculus?还有可不可以介绍一下关于Related rates的知识?我是要做英文的presentation.所以最好是英文.不过中文也可以,不需要太长.
Solution:
Let V, r and h be the volume,radius and height of the tree trunk respectively.
V = πr²h
dV/dt = 2πrh(dr/dt) + πr²(dh/dt)
= 2π×1.5×50×0.25 + π1.5²×1 = 39.75π
If we use algebraic method to solve it, we can find average rate of change, rather than instanteneous value. The volume of the tree trunk is simultaneously increased in all directions, algebraic method cannot be applied.
Let V, r and h be the volume,radius and height of the tree trunk respectively.
V = πr²h
dV/dt = 2πrh(dr/dt) + πr²(dh/dt)
= 2π×1.5×50×0.25 + π1.5²×1 = 39.75π
If we use algebraic method to solve it, we can find average rate of change, rather than instanteneous value. The volume of the tree trunk is simultaneously increased in all directions, algebraic method cannot be applied.