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 ([tex]V[/tex]), measured in cubic meters, is represented by the following formula:
[tex]V = \pi\cdot r^{2}\cdot h[/tex] (1)
Where:
[tex]r[/tex] - Radius, measured in meters.
[tex]h[/tex] - Height, measured in meters.
Then, we differentiate (1) in time:
[tex]\dot V = 2\pi \cdot r\cdot h \cdot \dot r +\pi \cdot r^{2}\cdot \dot h[/tex] (2)
Where:
[tex]\dot V[/tex] - Rate of change of the volume, measured in cubic meters per minute.
[tex]\dot r[/tex] - Rate of change of the radius, measured in meters per minute.
[tex]\dot h[/tex] - Rate of change of the height, measured in meters per minute.
Then, we clear the rate of change of the height:
[tex]\pi \cdot r^{2}\cdot \dot h = \dot V-2\pi \cdot r\cdot h \cdot \dot r[/tex]
[tex]\dot h = \frac{\dot V - 2\pi \cdot r\cdot h\cdot \dot r}{\pi \cdot r^{2}}[/tex]
[tex]\dot h = \frac{\dot V}{\pi \cdot r^{2}}-\frac{2\cdot h\cdot \dot r}{r}[/tex]
[tex]\dot h = \frac{1}{r}\cdot \left(\frac{\dot V}{\pi \cdot r}-2\cdot h\cdot \dot r \right)[/tex] (3)
If we know that [tex]\dot V = -3\,\frac{m^{3}}{min}[/tex], [tex]r = 7\,m[/tex], [tex]h = 4\,m[/tex] and [tex]\dot r = 0\,\frac{m}{min}[/tex], then the rate of change of the height is:
[tex]\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][/tex]
[tex]h = -0.019\,\frac{m}{min}[/tex]
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.
