Calculus Optimization Problem (please help!)

Question

There's an optimization problem for my AP Calculus class that I am stuck on.

You just adopted an iguana and are trying to build a pen with a rectangular bottom and 4 sides with the largest possible volume (essentially an open-top box). Using only 75 feet of building materials, what dimensions will produce a pen with maximum volume?

Here's the link to a diagram:

https://drive.google.com/file/d/1S7i27UKA_f4cvpPbUv8U-BGJRSmHWXzz/view?usp=sharing

Thank you!!

Answer 1

Hi Anna C.

Au contraire, Jacques D.!

Visualize this problem first as a "spherical" box: a sphere has the maximal volume for a particular dimensional framing (and also, for a given surface area, though this problem unaccountably doesn't attempt to restrain the iguana inside the box). However, unlike meshwork-sides and bottom for an top-open box, we are only concerned with the framing here. That is symmetrical all around the box; and a cube has the maximal volume of any linear dimensioned rectangular-prismatic shape. So just count the framing members (12), and divide the 75 ft of frame board equally amongst.

I'm still disturbed by the apparent procurement of the iguana before adequate facility is constructed for retention. Similarly, you should be prepared for the dietary needs of your new pet (among other needs), check out: http://poisonousplants.ansci.cornell.edu/toxiguana/iguana.html#Edible%20and%20Toxic .

-- Cheers, --Mr. d.

Answer 2

The problem is not clear. What are "feet" of building material? You also need another constraint on the amount of material as you have 3 independent variables (2 constraints and the derivative of volume = 0 in terms of the variable that is left allows for calculation).