## mariomintchev 3 years ago A conical salt spreader is spreading salt at a rate of 2 cubic feet per minute. The diameter of the base of the cone is 4 feet and the height of the cone is 5 feet. How fast is the height of the salt in the spreader decreasing when the height of the salt in the spreader (measured from the vertex of the cone upward) is 3 feet? Give your answer in feet per minute. (It will be a positive number since we use the word “decreasing”.)

1. mariomintchev

@AccessDenied @ash2326 @Chlorophyll @jagatuba @jhonyy9 @phi @TuringTest @Mertsj

2. TuringTest

always draw the pic...

3. TuringTest

|dw:1333043238193:dw|the radius of the whole cone is 2 and the height is 5

4. TuringTest

|dw:1333043327825:dw|at some given moment the cone of remaining salt is radius r and height h from this we can find similar triangles....

5. TuringTest

|dw:1333043429599:dw|so we get the proportion$\frac25=\frac rh\implies r=\frac25h$now we clearly need the formula for the volume of the cone, which is$V=\frac13\pi r^2h$to get this all in terms of one variable we will note the fact we got from the similar triangles....

6. TuringTest

$V(h)=\frac13\pi(\frac25h)^2h=\frac4{75}\pi h^3$can you differentiate this with respect to time?

7. mariomintchev

.44

8. TuringTest

I'm trying to figure out what you did here can you just show me what you get after differentiating wrt time?

9. mariomintchev

2*(75)/(4)*(3.14)*(27)=.44

10. TuringTest

don't plug in any numbers yet

11. mariomintchev

I'm using my notes. .44 is the correct answer.

12. TuringTest

ok, I'm not using a calculator, so I didn't know that I just wanted to make sure you knew how to differentiate this with respect to time

13. TuringTest

I don't suppose you feel like humoring me and showing me what you get after differentiating $$before$$ plugging in the numbers?

14. mariomintchev

yeah... h^3 becomes 3h^2 then 3(3)^2 =27

15. TuringTest

no that's not how you differentiate wrt $$time$$ you differentiated wrt to height, which makes this problem unsolvable

16. TuringTest

I mean, where did you get 0.44 aside from the fact that you already knew the answer? you forgot to differentiate $$implicitly$$ h, which must be done because it is a function of time (we want rate change of height in units of time, dh/dt)

17. mariomintchev

18. TuringTest

so then how did you get 0.44

19. mariomintchev

2*(75)/(4)*(3.14)*(27)=.44 i got it by doing this ^^^^

20. TuringTest

ok, see I couldn't decipher that you did it right, because you plugged in the numbers already, so my brain can't read the problem backwards that fast!

21. TuringTest

the way I was writing... so we have$V=\frac4{75}\pi h^3$differentiating with respect to $$time$$ we get${dV\over dt}=\frac4{25}\pi h^2{dh\over dt}$so we solve for dh/dt, which you did but when you plugged in the numbers it was hard for me to see that you had done that bit correctly

22. TuringTest

well, nice job then :D so for future reference: these cone problems are always done using the similar triangle trick that is sort of the motto here see ya 'round

23. mariomintchev

alrighty thanks :)