Frustrated on Old Assignment (a frustum of a right circular cone)

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RogersSteve

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Apr 11, 2018
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Hello!

My Calc 3 professor recently gave us an assignment. We had to find the optimal dimensions for a frustum shaped cup to minimize surface area and volume had to equal 8 ounces. I have attached the original assignment, which is long past due now. I have struggled with it back and forth for approximately 10 hours, went to tutoring, went to him for help, and haven't found anything. I'd just like to know where I went wrong. I can't upload a picture of my work, but I have done the following.

1) Found the volume and area equations (Area is without the top part, as it is a cup.)
2) Use LaGrange theorem to break up the equations using partial derivatives.
3) Attempted to solve a system for one variable
4) Solved for H, but wasn't able to substitute this into another solvable equation.

Please help, it's been driving me nuts for weeks!Screenshot 2018-04-11 11.16.24.jpg
 
Just realized the picture is pretty blurry. The text is as follows:

MA 243: Assignment 4
Due Friday, March 30


Q) A company manufactures paper cups that are designed to hold 8 fluid ounces
each. The cups are in the shape of a frustum of a right circular cone (so the top and
bottom of the cup are circles, not necessarily of the same size, and the side pro file is
that of a trapezoid). What are the dimensions for a paper cup that minimizes the
amount of material used? (Exact answer please.)

REMARKS
 Please ensure that you clearly and correctly explain what you're doing. If you
have any doubts about the clarity or correctness of your exposition, please ask!


 A correct mathematical setup is obviously essential for this problem. If the
setup is incorrect you will not earn any points for work done trying to solve the
problem. (And this assignment will be worth a lot of points.)


 Some of the algebra may be tricky, so all work past a certain point will count as
extra credit.
 
RogersSteve said:
...I have struggled with it back and forth for approximately 10 hours, went to tutoring, went to him for help, and haven't found anything. I'd just like to know where I went wrong.
Click to expand...
Unfortunately, it is not possible for us to trouble-shoot work that we cannot see.

Kindly please reply with a typed-out listing of your steps, or else nice clear images of your working. Thank you! ;)
 
Carloyn said:
I can't understand left side image. Can you please take another snap with clear zoom? So glad of you.
Click to expand...

Hi Carloyn! I'll be honest, the left side image is not important. That's just a scribbling where I set two equations equal to each other to help me think. It didn't go anywhere.
 
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