Answer:
The rate at which the water level is falling when the water is 4 meters deep to the nearest hundredth is 0.02 meters per minute.
Step-by-step explanation:
The volume of a cylinder (
), measured in cubic meters, is represented by the following formula:
(1)
Where:
- Radius, measured in meters.
- Height, measured in meters.
Then, we differentiate (1) in time:
(2)
Where:
- Rate of change of the volume, measured in cubic meters per minute.
- Rate of change of the radius, measured in meters per minute.
- Rate of change of the height, measured in meters per minute.
Then, we clear the rate of change of the height:
![\pi \cdot r^{2}\cdot \dot h = \dot V-2\pi \cdot r\cdot h \cdot \dot r](https://tex.z-dn.net/?f=%5Cpi%20%5Ccdot%20r%5E%7B2%7D%5Ccdot%20%5Cdot%20h%20%3D%20%5Cdot%20V-2%5Cpi%20%5Ccdot%20r%5Ccdot%20h%20%5Ccdot%20%5Cdot%20r)
![\dot h = \frac{\dot V - 2\pi \cdot r\cdot h\cdot \dot r}{\pi \cdot r^{2}}](https://tex.z-dn.net/?f=%5Cdot%20h%20%3D%20%5Cfrac%7B%5Cdot%20V%20-%202%5Cpi%20%5Ccdot%20r%5Ccdot%20h%5Ccdot%20%5Cdot%20r%7D%7B%5Cpi%20%5Ccdot%20r%5E%7B2%7D%7D)
![\dot h = \frac{\dot V}{\pi \cdot r^{2}}-\frac{2\cdot h\cdot \dot r}{r}](https://tex.z-dn.net/?f=%5Cdot%20h%20%3D%20%5Cfrac%7B%5Cdot%20V%7D%7B%5Cpi%20%5Ccdot%20r%5E%7B2%7D%7D-%5Cfrac%7B2%5Ccdot%20h%5Ccdot%20%5Cdot%20r%7D%7Br%7D)
(3)
If we know that
,
,
and
, then the rate of change of the height is:
![\dot h = \left(\frac{1}{7\,m} \right)\cdot \left[\frac{-3\,\frac{m^{3}}{min} }{\pi\cdot (7\,m)-2\cdot (4\,m)\cdot \left(0\,\frac{m}{min} \right)} \right]](https://tex.z-dn.net/?f=%5Cdot%20h%20%3D%20%5Cleft%28%5Cfrac%7B1%7D%7B7%5C%2Cm%7D%20%5Cright%29%5Ccdot%20%5Cleft%5B%5Cfrac%7B-3%5C%2C%5Cfrac%7Bm%5E%7B3%7D%7D%7Bmin%7D%20%7D%7B%5Cpi%5Ccdot%20%287%5C%2Cm%29-2%5Ccdot%20%284%5C%2Cm%29%5Ccdot%20%5Cleft%280%5C%2C%5Cfrac%7Bm%7D%7Bmin%7D%20%5Cright%29%7D%20%5Cright%5D)
![h = -0.019\,\frac{m}{min}](https://tex.z-dn.net/?f=h%20%3D%20-0.019%5C%2C%5Cfrac%7Bm%7D%7Bmin%7D)
The rate at which the water level is falling when the water is 4 meters deep to the nearest hundredth is 0.02 meters per minute.