AP Calc Project: Find Dimensions of 12 Fl. Oz. Cola Can

  • Thread starter Thread starter Vigo
  • Start date Start date
  • Tags Tags
    Project
Click For Summary

Homework Help Overview

The problem involves designing a right circular cylinder to hold 12 fluid ounces of a soft drink while minimizing the amount of material used in its construction. The context is rooted in applications of derivatives in calculus.

Discussion Character

  • Exploratory, Mathematical reasoning, Problem interpretation

Approaches and Questions Raised

  • Participants discuss translating the physical problem into mathematical terms, focusing on minimizing surface area under a volume constraint. There are attempts to express height in terms of radius and to derive a function for surface area. Questions arise regarding the correctness of the expressions and the relevance of the volume conversion factor.

Discussion Status

The discussion is active, with participants sharing their interpretations and mathematical expressions. Some guidance has been offered regarding the optimization process, but there is no explicit consensus on the correctness of the approaches or the final interpretation of the results.

Contextual Notes

Participants are working under the constraint of using a specific volume measurement and are questioning the implications of unit conversions in their calculations.

Vigo
Messages
21
Reaction score
0
In my AP Caluculus class we are on the chapter of derivative applications and we have this project to di.

A right circular cylinder is to be designed to hold 12 fluid ounces of a soft drink and to use a minimum amount of material in construction. Find the required dimensions for the container. (1 fl. oz. = 1.80469 inches cubed)

If someone could please help to get me started on this problem I would be very grateful.
 
Physics news on Phys.org
Vigo said:
In my AP Caluculus class we are on the chapter of derivative applications and we have this project to di.
A right circular cylinder is to be designed to hold 12 fluid ounces of a soft drink and to use a minimum amount of material in construction. Find the required dimensions for the container. (1 fl. oz. = 1.80469 inches cubed)
If someone could please help to get me started on this problem I would be very grateful.
[tex]\text{SA}=2\pi r\left(h+r\right)[/tex]

[tex]\text{V}=\pi r^{2}h=12[/tex]

I suggest you find an expression for h in terms of r in the second and plug it into the first to simplify. The rest should be easy assuming you know how to optimize a function using differential calculus.
 
Translate the problem into a mathematical problem.
You want to minimize the surface area of a cylinder under the constraint that the volume must be some given volume V.

So start by drawing a picture. Introduce variables that will be important (height, radius, suface area). And relations between them. This is the given. What is the unknown?
 
OK so:

h = 12/(pi*r^2)

and

f(x) = (2*pi*r)*(12/pi*r^2) + 2(pi*r^2)

Find the derivative of f(x) and set that equal to 0.

Is all of this right?
If it is, what does the final answer tell you?
And where does the 1 fl. oz. = 1.80469 inches cubed come into this problem?
Thanks again.
 
Okay, so f is the surface area of the can as a function of the radius r (not x!). You can minimize this using whatever knowlegde you have of calculus (=yes, differentiating would be the standard procedure).

The "1 fl. oz. = 1.80469 inches cubed" comes from the conversion from some archaic unit such as fluid ounce to another one, namely the inch^3. You should use whatever dimensions you are comfy with and do the proper conversion.