Answer:
The rate of change for the height of the cone is [tex]-2.849\times 10^{-3}[/tex] centimeters per minute.
Step-by-step explanation:
We derive an expression for the rate of change for the height of the cone by differentiating the volume formula given on statement:
[tex]\dot V = \frac{\pi}{3}\cdot (2\cdot r\cdot h \cdot \dot r + r^{2}\cdot \dot h)[/tex] (1)
Where:
[tex]\dot V[/tex] - Rate of change for the volume of the cone, measured in cubic centimeters per minute.
[tex]r[/tex] - Radius of the cone, measured in centimeters.
[tex]h[/tex] - Height of the cone, measured in centimeters.
[tex]\dot r[/tex] - Rate of change for the radius of the cone, measured in centimeters per minute.
[tex]\dot h[/tex] - Rate of change for the height of the cone, measured in centimeters per minute.
If we know that [tex]\dot V = -148\,\frac{cm^{2}}{min}[/tex], [tex]r = 222\,cm[/tex], [tex]V = 21\,cm^{3}[/tex], [tex]\dot r = -5\,\frac{cm}{min}[/tex], then the rate of change for the height is:
[tex]h = \frac{3\cdot V}{\pi\cdot r^{2}}[/tex] (2)
Where [tex]V[/tex] is the volume of the cone, measured in cubic centimeters.
[tex]h = \frac{3\cdot (21\,cm^{3})}{\pi\cdot (222\,cm)^{2}}[/tex]
[tex]h \approx 4.069\times 10^{-4}\,cm[/tex]
From (1):
[tex]\frac{3\cdot \dot V}{\pi} = 2\cdot r\cdot h\cdot \dot r + r^{2}\cdot \dot h[/tex]
[tex]r^{2}\cdot \dot h = \frac{3\cdot \dot V}{\pi}-2\cdot r\cdot h\cdot \dot r[/tex]
[tex]\dot h = \frac{3\cdot \dot V}{\pi\cdot r^{2}}-\frac{2\cdot h\cdot \dot r}{r}[/tex]
[tex]\dot h = \frac{3\cdot \left(-148\,\frac{cm^{3}}{min} \right)}{\pi\cdot (222\,cm)^{2}} -\frac{2\cdot (4.069\times 10^{-4}\,cm)\cdot \left(-5\,\frac{cm}{min} \right)}{222\,cm}[/tex]
[tex]\dot h = -2.849\times 10^{-3}\,\frac{cm}{min}[/tex]
The rate of change for the height of the cone is [tex]-2.849\times 10^{-3}[/tex] centimeters per minute.
