A microwaveable cup-of-soup package needs to be constructed in the shape of cylinder to hold 350 cubic centimeters of soup. The sides and bottom of the container will be made of styrofoam costing 0.03 cents per square centimeter. The top will be made of glued paper, costing 0.08 cents per square centimeter. Find the dimensions for the package that will minimize production costs.
Context: I'm a tutor and the student asked me this question. The student's website lists the answers as r = 3.12, h = 11.44 and minimum cost = 10.09, with no explanation given.
My work follows here.
| Cost | Volume |
| — | — |
| $$ Cost = 0.03 \pi r^2 + 0.08\pi r^2 +0.03\pi r h$$ | $$350 = V = \pi r^2 h$$ |
| | $$h = \frac{350}{\pi r^2} $$ |
| $$C = 0.11 \pi r^2 + \frac{10.5}{r}$$ | |
My answers are r = 2.48 cm, h = 19.32 cm and minimum cost = 6.36 cents
https://www.wolframalpha.com/input/?i=minimize+%28.11pi+r%5E2+%2B+0.03*350+%2Fr%29
Best Answer
The area of the side is $2\pi r h$, not $\pi r h$.
This makes the answer https://www.wolframalpha.com/input/?i=minimize+%28.11pi+r%5E2+%2B2*+0.03*350+%2Fr%29
r= 3.12044->3.12, h= 11.441593->11.44, and the full cost is 10.09 as the website expected.