## zeesbrat3 one year ago Water is drained out of tank, shaped as an inverted right circular cone that has a radius of 6cm and a height of 12cm, at the rate of 3 cm3/min. At what rate is the depth of the water changing at the instant when the water in the tank is 9 cm deep? Give an exact answer showing all work and include units in your answer.

1. zeesbrat3

@jim_thompson5910

2. jim_thompson5910

What do you have so far?

3. zeesbrat3

r = 6cm h = 12cm r = 3cm^3/min $v = \pi r^2h$

4. jim_thompson5910

r = 3cm^3/min is incorrect I think you meant to say dV/dt = -3 to represent the fact that the volume is decreasing by 3 cubic cm

5. jim_thompson5910

also, you have the wrong volume formula

6. jim_thompson5910

volume of a cone $\Large V = \frac{1}{3}\pi*r^2*h$

7. zeesbrat3

Oh... So then you take the derivative, yes?

8. jim_thompson5910

yeah but first you have to somehow get everything in terms of h you have to find a connection between r and h and do a substitution

9. jim_thompson5910

a picture might help |dw:1437352686967:dw|

10. zeesbrat3

r = 1/2h?

11. jim_thompson5910

|dw:1437352732409:dw|

12. jim_thompson5910

yeah that looks good, r = h/2

13. jim_thompson5910

plug that in, derive, then isolate dh/dt

14. zeesbrat3

$\frac{ 2 }{ 3 } \pi (\frac{ h^2 }{ 2 })$

15. jim_thompson5910

When you plugged r = h/2, and simplified, did you get $\Large V = \frac{1}{12}\pi h^3$ ??

16. zeesbrat3

So you get $\frac{ dv }{ dt } = \frac{ 1 }{ 4 } \pi h^2 \frac{ dh }{ dt }$

17. jim_thompson5910

good

18. jim_thompson5910

dv/dt = -3 h = 9 solve for dh/dt

19. zeesbrat3

$-3 = \frac{ 1 }{ 4 } \pi (9)^2 \frac{ dh }{ dt }$ $\frac{ -12 }{ 81 \pi } = \frac{ dh }{ dt }$

20. jim_thompson5910

make sure you reduce the fraction as much as possible

21. zeesbrat3

$\frac{ -4 }{ 27 \pi } = \frac{ dh }{ dt }$

22. jim_thompson5910

yep $\Large \frac{dh}{dt} = -\frac{4}{27\pi} \approx -0.047157$ the units for dh/dt are cm/min the rate of the depth of the water is changing at roughly -0.047157 cm/min at exactly the depth of h = 9 cm

23. zeesbrat3

Thank you!!

24. jim_thompson5910