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

[Vigo]

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.

 

[amcavoy]

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.

$$\text{SA}=2\pi r\left(h+r\right)$$

$$\text{V}=\pi r^{2}h=12$$

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.

 

[Galileo]

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?

 

[Vigo]

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.

 

[Galileo]

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.