## anonymous 4 years ago Need help with a related rates calculus problem! I'll type it!

1. anonymous

A candy company needs a custom box for their truffles. The box they've chosen is in the shape of a cylinder with a hemisphere of the same radius on top. The total volume of the box is V= 1/2(4r^2pi/3) + r^2pi(y-r), where y is the height of the box and r is the radius. Originally the candy box was designed to have a height of 6 inches and a radius of 2, but the shipper suggests that the boxes be made slightly shorter. You now need to adjust the radius so that the height is reduced to 5.75 but the volume remains constant. A. Find the value of dr/dy at the point r =2, r = 6. Help with process

2. anonymous

Just need help getting started.

3. Hero

mertsj would be surprised if I solved this

4. anonymous

Why is that? I really don't understand how dy/dr can be taken on that formula. I'll write it out nicer if you like.

5. anonymous

$1/2(4\pi r^2\div 3)+\pi r^2(y-r)=V$

6. Hero

Okay, I got it

7. anonymous

Thanks a lot. Let me know if you need anything else.

8. Hero

r = 2, y = 6, right?

9. anonymous

Yes.

10. anonymous

11. Hero

-3/11 = dr/dy

12. anonymous

Hero, thank you, but this is free response and I needed help with the process.

13. anonymous

Can anyone help me with the process?

14. Hero

Maybe. Do you know how to take the derivative of Volume?

15. anonymous

Yes. Typically though, it is d/dt. That is what is tripping me up. This time it is dr/dy. What changes?

16. Hero

derivative of the radius with respect to y, rather than t

17. anonymous

So, when you end up taking the derivative of an 'y' values, you get dr/dy? ex. dr/dy(2y^2) = 4y dr/dy?

18. anonymous

I suppose my question is this: What happens mathematically when you take the derivative of radius in respect to height?

19. Hero

Well, it doesn't exactly work like that but you're on the right track. If you have an expression with y and you take the derivative of r with respect to y, you take the derivative normally, then tack on the dr/dy

20. anonymous

Okay. That _doesn't_ happen with 'r' terms though.

21. Hero

You take the derivative with r as normal. The derivative of r with respect to y only applies to the y variable

22. anonymous

Thank you! That's what I needed to know.

23. Hero

Tell mertsj that I helped you. When you get the answer, let me know. I probably shouldn't have told you so early.

24. Hero

You do realize that finding dr/dy is only the first part. The second part is, you have to replace y = 5.75 to find r while keeping the volume constant

25. anonymous

Honestly, I don't know how to tell him if he's not online, but if I see him chatting I will; I'm fairly new here. I will let you know when I get the answer. Thank you, and yes, I figured that part out. It was the implicit stuff that was killing me.

26. Hero

27. anonymous

And thank you for your time :)

28. Hero

Actually, we can go to a site called vyew

29. Hero

Then I can watch you do the steps

30. anonymous

Uhh, is that...video chat? Because I think I'll pass for the afternoon, thanks.

31. anonymous

I'm very close to the answer, just give me another minute or two.

32. Hero

It's not video chat. Why does everyone automatically assume video chat?

33. anonymous

Might have to do with the 'watch' part.

34. Hero

It's just a site with an online whiteboard

35. Hero
36. anonymous

Okay. I've got a tablet. My handwriting should be decent.

37. Hero

Yes, your handwriting was very nice. I figured you must have a tablet

38. anonymous

Sorry, man. Let me keep working and see if I can figure out what I did wrong. I don't know why my microphone wouldn't work.

39. Hero

Well, I think it would be better if you worked on the mic, but okay

40. anonymous

I can keep trying.

41. anonymous

I got -1/5 this time. I'm checking wolframalpha.

42. Hero