Click to enlarge blackboard.
"Math is hard!"
........-- Talking Barbie
if u have an extremely rare Talking Barbie, Do Not Take It Out Of Its Box, store in Safe Deposit Box, wait. Eventually you can sell it and be Rich.
(Also Barbie's Very Much Younger Australian Surfer bf, Blaine, for whom she ditched wrinkled old Ken. But just for a few months, then she went back to wrinkled old Ken, and Mattel stopped manufacturing Blaine the Surfer Boy.)
When you take the Zeta Beam to the Akira Kurasawa ZetaDrome in Ciudad Vleeptron, the first place you'll want to go via free jitney to CV UnderWay is Fudge Tunnel station, right next to the fabulous Club Drek, whose music acts do not discriminate between Living Artists & Dead Musicians.
But every tourist to CV wants to take a selfie at The Great Halvah Pyramid (nearest stops: Greune, Rue Jarry), one of the Seven Wonders of Galaxy Dwingeloo-2. This awesome pyramid is constructed entirely of extremely thin squares of halvah.
Above, only slightly modified, is some Crummy Old Vleeptron Wine: Central American Rockwell's Solution to the ancient mystery of the Volume of any pyramid made of very thin slices of halvah.
(Central American Rockwell is Vleeptron's and Planet Mongo's Prime Contractor in Deep Space & Intergalactic activities.)
How Old is this Crummy Wine? Well, here's a clue -- the 1-1 Final Score by Team USA vs. Italy in semifinals of World Cup Football ... which was particularly amazing because Team USA had only just learned the Rules of this alien un-American game 1 month earlier.
What follows is all just 1st Year integral calculus, a Walk in the Park for any female (except Talking Barbie) or any male willing to wake at 4 am, study every page, do every problem (including Optionals), and go to bed the next 2 am to get a passing grade on the midterm exam.
Follow it step by step until you come to The Miracle at the bottom. Say "Wow!" and continue on to The Halvah Pyramid Volume Solution.
Earthers think the Turks invented halvah, but actually halvah was invented by the vanished jungle people of Planet Hoon, the Vodeeodo. I brought some back on the Zeta Beam, then travelled back in time in my Heathkit TM-212, and gave it to a Turkish guy in 1050 AD.
The Vodeeodo considered Halvah a Sacred Food, and only Royalty and the Priests were allowed to eat it. If they caught an Ordinary Guy eating halvah, they would immediately seize and disembowel him, and eat any undissolved halvah they found in his intestines.
They made halvah into square sheets of such incredible thinness that when you put a piece on your tongue, Taste was the only sensation -- your tongue and teeth could detect no mass or bulk, there was no chewing required, it just melted into deliciousness on your tongue.
Eventually they began building their Sacred Pyramids by stacking gazillions of incredibly thin halvah sheets one on top of the other. This Fairly Small Flat-Top Pyramid is the only one left; travellers and tourists long ago ate all the other Pyramids.
Given such a Pyramid built of incredibly thin square halvah sheets, how can we derive the Volume Formula which won Jim the pizza slices?
If the LENGTH of a side of each square sheet is s units, then its AREA is s² units squared.
Each sheet is so incredibly thin that it's impossible to assign a number to its thickness, so we just call the thickness dx .
Make no mistake -- dx IS really a dimension, it's Not Nothing. But if we insisted on assigning it a number like 0.000000000000000002 units, it would be cumbersome and unwieldy -- and still too large a number to accurately express the incredible thinness of a halvah slice.
So the VOLUME of each sheet of halvah is
s² x dx
s² dx units cubed
h = the height of a Pyramid
b = the length of a side of its square base
a = the length of a side of its square roof
we want to derive the simple Volume Formula
V = (h/3)(a² + ab + b²)
Step 1. Kick the Pyramid and knock it over on its side.
Step 2. Skewer the centers of the bottom and top halvah sheets with a long, incredibly thin knitting needle. If you want to call this knitting needle the x-Axis, that's okay with me and Rene Descartes.
Now we want to juggle the measured lengths a, b and h to find the equation
y = f(x)
for the straight line from the midpoint of the side of the base to the midpoint of the side of the roof.
With this equation, no matter which halvah sheet we choose, we'll know exactly what the length of a side of it is at that height of the Pyramid: 2y units
And that sheet's AREA will be (2y)² = 4y² units squared
and that sheet's VOLUME will be 4y² dx units cubed.
If we could Sum the Volumes of All These halvah sheets, we'd know the VOLUME of the whole Pyramid.
That's what that screwy Integral Symbol is -- a Sum. That's why Leibniz made it look like a stretched out S.
We're going to Sum the Volumes of all halvah sheets
from the Pyramid's Base (x = 0) to its Roof (x = h) .
Get a Big Bottle of Extra Strength Aspirin. (Canadians may use Over-The-Counter Aspirin with Halvah. Americans need a doctor's prescription for all medications containing Halvah.)
The y equation isn't too hard to figure out. Things start to get algebraically jiggy when we multiply it by itself to find y² . Now take the first 3 aspirin.
Down by the Smiley Face, we Evaluate The Definite Integral.
The Face Is Smiling because the Lower Limit x = 0 . Because each of the 3 terms contains a factor x , and 0 times anything = 0 , this effectively halves the Algebra Nightmare we have to endure. We only have to evaluate this nasty expression for the Upper Limit x = h .
Four lines below, A Wonderful Algebra Miracle Happens, and the whole verkakte mess collapses nicely into the simple Volume formula. Please do not eat the Pyramid.
P.S. If you don't kick the Pyramid over on its side, then you end up with
x = f(y)
which is perverted and Against The Law, and I don't know how to evaluate stuff like that.
posted by Vleeptron Dude | 19:41
Blogger Abbas Halai said...
halva in my language and culture has been synonymized with any sweet dessert/confectionery.
hmmm well wait, now that i think about it, it hasn't but theres a very specific type of sweet that its synonymous with and i don't know the english translation for it. i'm not a big fan of most halva's but i do have a favourite. its barfi that you get in rawalpindi.
Blogger Bob Merkin said...
okay soon as i get thru customs at Pindi Airport, I buy the barfi, eat it -- and then promptly drop dead from diabetic sugar overload. that halvah stuff man that is so sweet that my endocrinologist doesn't even put it on the Forbidden Foods list, but lists it on another page: How A Diabetic Can Commit Suicide.
(But maybe if I can find an incredibly thin sheet of barfi, just dx units thick, I'll give it a try. How much harm can that do?)
DID YOU SEE MY REVISED IMAGE? It's my Homage to TEAM USA, which is still Mathematically Alive in the World Cup after tying Italy the other day. "Mathematically Alive" is a charitable way that the ESPN World Cup commentators can remain in Reality Denial for a few more days.
The Agence-Vleeptron Presse Sports Team will shortly have more to say about Team USA, which Americans pay about as much attention to as they pay to Vleeptron's free calculus lessons.
But the guy making that super kick in the game against Italy is named McBride or MacBride. He might even be an American, who knows, stranger things have happened. I will check this out. In the stands is a largely nude fan who painted himself as our Flag.