## zeesbrat3 one year ago A water filter has the shape of an inverted right circular cone with base radius 3 meters and height 9 meters. Water is being pumped into the filter at the rate of 10 meters3/sec. Find the rate, in meters/sec, at which the water level is rising when the water is 3 meters deep.

1. zeesbrat3

Hey

2. ganeshie8

Hey! related rates problem

3. ganeshie8

|dw:1436729337203:dw|

4. zeesbrat3

Ya, my favorite...

5. ganeshie8

|dw:1436729438553:dw|

6. zeesbrat3

What's the next step?

7. Astrophysics

I miss these problems, so fun. What we're given is $\frac{ dV }{ dt } = 10\frac{ m^3 }{ s }$ and you have to find $\frac{ dh }{ dt }$ ganeshie used similar triangles above to find r so we can use the volume formula $V = \frac{ 1 }{ 3 }\pi \left( \frac{ h }{ 3 } \right)^2h = \frac{ 1 }{ 9 }\pi h^3$ now we differentiate!!

8. zeesbrat3

So.. $\frac{ dv }{dt? } = \frac{ 1 }{ 3 }pih^2$

9. Astrophysics

$\frac{ dV }{ dt } = \frac{ \pi }{ 3 }h^2 \frac{ dh }{ dt }$ notice we differentiate respect to t

10. Astrophysics

Remember we are using the chain rule here, sorry @ganeshie8 for intruding hehe I just like these problems :D

11. Astrophysics

Now we solve for $\frac{ dh }{ dt }$ can you go ahead and try that please

12. Michele_Laino

here is my reasoning: |dw:1436729996057:dw| we can write: $\Large \begin{gathered} dV = \pi {r^2}dh \hfill \\ \hfill \\ \frac{{dV}}{{dt}} = \pi {r^2}\frac{{dh}}{{dt}} \hfill \\ \hfill \\ \frac{{dV}}{{dt}} = \pi {\left( {\frac{h}{3}} \right)^2}\frac{{dh}}{{dt}} \hfill \\ \hfill \\ \frac{{dV}}{{dt}} = \frac{{\pi {h^2}}}{9}\frac{{dh}}{{dt}} \hfill \\ \hfill \\ \frac{{dV}}{{dt}} = \frac{\pi }{{27}}\frac{{d\left( {{h^3}} \right)}}{{dt}} \hfill \\ \end{gathered}$

13. zeesbrat3

$10 = \frac{ 1 }{ 3 }\pi(9)^2\frac{ dh }{ dt }$

14. Astrophysics

$\frac{ dh }{ dt } = \frac{ 3 }{ \pi h ^2 } \frac{ dV }{ dt }$

15. Astrophysics

$\frac{ dh }{ dt } = \frac{ 3 }{ \pi(3)^2 } \times 10 = \frac{ 30 }{ 9 \pi ^2 }$

16. Astrophysics

That is the rate of water rising

17. zeesbrat3

So $\frac{ 30 }{ 88.7364 }$

18. zeesbrat3

Or .33807

19. Astrophysics

The second one

20. anonymous

This is wrong

21. anonymous

The answer is 3.18. Astrophysics is wrong on so many levels.

22. Astrophysics

@dahmanman show your work and I might believe you.