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Re: When math makes sense  w/ cooking, consruction
Posted:
May 29, 2013 2:03 AM


You are, however, oversimplifying realworld problems (as is almost always the case so that developing genuine problem solving skills is possible  bullet trajectories without air resistance?). For example, making other shapes out of strips of paper adds a huge (time wasting?) problem to rolling up a rectangle or cutting corners out of one, maybe saving a rectangle for the top so the problems are comparable. And it isn't just kids in school, slipform grain silos may not be mathematically efficient but both construction and grain stored per unit land area are vastly superior to big balloons or some such "ideal" shape. Even the ubiquitous water towers that I grew up with in the flat Midwest that were balloons have been augmented  if not replaced  by more efficient silotype construction. And your big lakes have problems as well, both mathematical (there is a mathematical optimal) and practical considerations such as evaporation and human (agriculture, cities, and environmental  Hetch Hetchy remains controversial a century later and, even then, it took the San Francisco earthquake and fires to trump John Muir and get permission).
All these things are critically important but not wasting zerosum school math time. That's math avoidance, not math teaching.
Wayne
At 04:54 PM 5/28/2013, kirby urner wrote: >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.



