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Re: When math makes sense  w/ cooking, consruction
Posted:
May 28, 2013 7:54 PM


Taking a more algebraic approach, using the tools I'd expect in a STEM class for the age of the student in the video, I find that the best strategy for maximizing volume is to maximize the radius and minimize the height.
If you have a fixed surface area equal to 8.5 x 11 inches, or 93.5 square inches, then cut very narrow strips and splice them together to make as giant a circle as you can.
Looking at the formula, this makes sense because w and h are inversely proportional and h * w always multiplies to the same answer. However, if we make r a function of width and V = pi * radius * radius * height then clearly investing in radius is best, as it's a 2nd power payout, whereas investing in height is less bang for the buck.
A height of 92 inches means a tube with a radius of 0.16 inches, a circumference of just a tiny bit over an inch.
The volume will be only 7.56 cubic inches whereas if you have a height of 0.1 inches your width will be 935 inches (almost 78 feet around) and your volume will be a whopping 6956.9 cubic inches, which is huge.
A vast circular shallow lake is the way to go, versus a tall thin tubular straw. The latter maximizes surface to volume, whereas the former minimizes it.
Kirby
You'll need to view the plaintext version of this post in the archives (button upper right) if you want this code to be indented properly; Python has significant whitespace meaning indentation is used when parsing. The table also looks better when the tabs (\t) are not ignored (HTML suppresses whitespace by design).
Note that Width is defined as the circumference of the cylinder, as if unrolled to make a rectangular sheet.
from math import pi
def cylvol(height, radius): return pi * radius**2 * height
def get_width(height): return (8.5)*(11)/height
def get_radius(h): w = get_width(h) return w/(2*pi)
print("Paper surface of 8.5 x 11 =", 8.5 * 11, "square inches")
print("Height:","Width:","Radius:","Surface:","Vol:", sep="\t") for h in [ 0.1, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11.4, 11.49, 92 ]: r = get_radius(h) w = get_width(h) columns = "{height:>6.2f}\t{width:>6.2f}\t{radius:>6.2f}\t{surface:6.2f}\t{volume:>8.2f}" print(columns.format(height=h,width=w,radius=r,surface=h*w,volume=cylvol(h,r)))
Running the above:
Paper surface of 8.5 x 11 = 93.5 square inches Height: Width: Radius: Surface: Vol: 0.10 935.00 148.81 93.50 6956.86 1.00 93.50 14.88 93.50 695.69 2.00 46.75 7.44 93.50 347.84 3.00 31.17 4.96 93.50 231.90 4.00 23.38 3.72 93.50 173.92 5.00 18.70 2.98 93.50 139.14 6.00 15.58 2.48 93.50 115.95 7.00 13.36 2.13 93.50 99.38 8.00 11.69 1.86 93.50 86.96 9.00 10.39 1.65 93.50 77.30 10.00 9.35 1.49 93.50 69.57 11.40 8.20 1.31 93.50 61.03 11.49 8.14 1.30 93.50 60.55 92.00 1.02 0.16 93.50 7.56
PS: if you're a high school student reading this and your STEM classes do not give you access to free software such as Python, then consider writing to the governor of your state and expressing displeasure at the backwardness of your curriculum and the unwanted negative impact on your future this deficiency may have. Demand to have an interactive full featured programming language in your curriculum, not just calculators. Don't take no for an answer. The hardware is dirt cheap and the software is gratis. Schools that can't supply such minimal equipment are poorly funded and administered. You deserve better. Have more self esteem.
It's not that I think governors should really be involved, but they've insisted in recent chapters. Also scan these remarks and see if the content looks at all familiar (a post in this same thread):
http://mathforum.org/kb/message.jspa?messageID=9128009
If not, more warning flags. Talk to your parents. Is distance education possible? Sometimes neighboring states have more enlightened options, or even far away states. Earn academic credit off shore or overseas if need be, using your computer as a communications device.
Do not settle for what just happens to be local, if it also happens to be inferior.



