GingerAle

I don’t think Lancelot was too disappointed by his loss.  On Election Day, he was distracted by a malfunctioning PORV (whatever that is) on a used, portable, nuclear reactor that he bought to power his banana and peanut processing plant.  His board of directors made wagers on whether Lancelot or the reactor would melt down first. 

To pay for the replacement parts, Lancelot sold “glow-in-the-dark bananas” and retooled the production line to make Manhattans. He called it “The Manhattan Project.” He also smuggled mangos across state boarders. The authorities caught him during his final run. Five weeks later, LL appended his mango-smuggling fiasco to a post about random probabilities of typing-monkeys. 

Everyone has a price.

This animated graphic is my favorite.

This gives an idea of the light and shadow and the knowledge and imagination of the science and mathematics that is Euler.

Your intelligence  or  Miss Melody’s?

You really are a ninja. You see the invisible probabilities for the chance of rain and Sheila’s weighted decisions. Amazing!!

Actually, in this case

\((-1^.5)^2\)

Is the same as

\(\left ( \sqrt{-1} \right )^2 \\ \text {is the same as } \sqrt{-1}= i \rightarrow i^2 = -1\\\)

I may have created more ambiguity with my post.

I checked the forum’s professional registry; none of the engineers on here are civil.

Most of them are grumpy and have bad attitudes.

Asinus is right, this is debated quite often in academic circles –at least among the science and math majors.

I’ll add another two cents worth.

A fraction is ostensibly the same as division.  Most will agree that all of these are the same thing the same as  

\(3 \div 5 \\ \dfrac {3}{5}\\ 3/5 \\\)

Most will also know that order of operations (PEMDAS) says division take precedence over addition. Meaning 5+20/4+1 = 11 (not 5)

But

\(\dfrac {5+20}{4+1} = 5\)

There are no parentheses here, and addition now appears to takes precedence over division --both before and after the division  to give the "correct" answer of 5.

(Take note, guest #7, you listed several calculators but left out this site’s calculator)

These are comments from my favorite troll:

If you paste this  48/(2)(9+3)  into the site calculator, it returns 2 as the solution.

This is isn’t a bug. The parentheses cause the value to be treated as a variable. Implicit multiplication of variables takes precedents over division – a noted exception, dating back to the late 1960s, to the normal convention of mathematical hierarchy . Herr Massow’s calculator is the only one I know of that that does this, and it’s probably because it allows the use of variables.

Of course, if you put the explicit multiplication operator in: 48/(2)*(9+3) or leave the parentheses off the 2 then it returns 288,  the normal solution for numeric values.

This calc does have one hierarchal fault. “Stacked powers” are right-associated. The calc resolves them from the left.

4^3^2 returns (4^3)^2 = 4096 (It shows the order at the top).

With the correct hierarchy, it is 262144.

It’s always a good idea to know the quirks and faults of the tools you might use.  I drove an old car once that ran great, but the breaks didn’t work so well –I had to drag my foot to help it stop. It sure wore out my shoes faster than normal. :)

Well, Mr. Banker, I thought you might like to know I was playing monopoly with my dog and (cheating) cat, when this question popped up.  I’ve seen this question before. A student in my Trig study group presented it as a practice question. We all took a stab at it, but none of us knew how to solve it. The student who presented it also had the solution for it, which is near verbatim to what you presented.  There is one major problem: the solution is wrong. 

I knew it was wrong, not because I am a math whiz –as I said in a previous post, that’s not on my résumé yet, and it certainly wasn’t then. I knew it was wrong because I have an intuitive understanding and a well-developed skill for art – including three-dimensional art.  Moreover, and most importantly, I’ve dipped into more than a few cheese bàlls in my life. Cheese bàlls are the only spherical-cheeses I’ve ever seen. I’ve seen many circular cylinders of cheese, including one of six feet in diameter and weighing 1323 lbs., displayed nine years ago in New York City’s Grand Central Market. It was Gouda from Holland. 

When I saw your solution, I thought, “Is this the banker? The banker doesn’t show work product, except for simple TVM interest rate problems.”  You post as a guest, so it’s not easy to tell. I decided you are the banker for a few reasons.

First, there was your introduction:

I hope you can follow this. You certainly can if you draw up the picture:

Sorry, I couldn't upload the picture.

Then your wrong (and PLAGERIZED) answer kind of gave it away. It was nearly verbatim to the one present by the student in my trig study group. The only difference was the last line gave the approximation of pi as (22/7) which gives the answer as exactly 50lbs. 

Finally, you followed your solution with a confirmation of plagiarism: 

I think I got it right!!.

It’s always a good idea to make sure the answer you are copying is the correct answer, because you are less likely to get caught (most of us learn that in grade school). In this case, the presentation is too unique to be duplicated by random chance.  In case you do not know what plagiarism is, it’s analogous to embezzlement. I’m sure you understand embezzlement. I figured out this is why you haven’t taken my suggestion to choose a user name.

http://web2.0calc.com/questions/hey-guys-i-posted-a-few-questions-and-i-can-t-find-them-anywhere-i-posted-it-yesterday-but-it-s-gone-what-do-i-do

it’s also why you cannot upload images.  You have restricted access where you are  I understand, it’s a country club by comparison but there are limits. 

As I said above, I was playing monopoly with my dog and cat. Normally it takes something important to pull me away from these games. However, it was the cat’s turn to be the banker and the dog always insists on the right to do a bank audit at the end, and, at a time of his choosing, before the end of the game.  It’s easy to understand why: the cat not only cheats, he embezzles when it’s his turn to be the banker.  He can’t help himself. (I’m thinking, you’d probably like my cat).   About the time you posted your reply to Sir Cphill’s answer, the dog called for an audit. This is never pretty, but it’s usually entertaining. 

In their younger years they use to “duke-it-out.”  They still do, sometimes, but it’s usually a short-lived battle. The dog will snarl at the cat, while the cat hisses and extends his claws. The dog will lunge at the cat but pivot his rump around as the cat swipes with his extended claws. The dog will yelp, as his moderately massive rump plows into the cat, sending him flying across the slippery floor, at which point the cat jumps on top of the china cabinet and waits for the dog to cool off. 

Usually, by the time I correct the cash discrepancies and fine the cat 20% of his winnings, the dog will have cooled off enough to let the cat return to the game, so they can trade-in their winnings for Scooby-Snacks and catnip.  Not this time, though.  He decided to keep the cat on his china cabinet “cell” for a while. Actually, they both just fell asleep -- they are elderly: seventeen for the cat, and fourteen for the dog. 

While the dog and cat were napping, I searched for my notes where my mentor explained the correct solution to me.  My mentor’s explanation clearly shows why the solution you (and the book) present is wrong.

Hi Ginger,

You are right,  the answer is BS. (The author may have smoked  a doobie before constructing this solution.  In fairness to the author, it would work if it was a circular cylinder.

This solution is incorrect, because the h(eight) is both subtracted from the radius and then squared and multiplied by it. This means the volume is not uniformly proportional over the height.  The only time a spherical segment of one base (spherical cap) is proportional to the other is when it’s divided in half.

This formula will give the volume for a spherical cap

\(V_{cap} = \dfrac{3}{4}*(\dfrac{h}{r})^{2} - \dfrac{1}{4}*(\dfrac{h} {r})^{3}*\dfrac{4}{3}(\pi r^{3}) \)

This is derived from an integration of Gauss’s Divergence Theorem

\(V_{cap}=\int_{cap} V = \int_{cap}{1 \over 3}\,\nabla\cdot\vec{r}\, V \int_{sur} \vec{r} \cdot \vec{s} \)

The easiest method to use for solving the spherical cap volume is the formula using the radius and height.

\(\displaystyle V_{cap} = \dfrac{1}{3}\pi h^2 (3r - h)\\ \)

The height can be found using

\(\text {Cap height (h)} = (r) \left( 1- (\cos\frac{\alpha}{2}) \right)\)

The radius is 1 (unit circle) and it’s 90 degrees for the angle corresponding to the chord length of sqrt(2). Divide the volume of the sphere (v=4/3pi*r^3 or 4.1888 for unit radius), by the volume of the cap.

\((1)* \left (1 - \cos \left ( \dfrac {90}{2} \right ) \right ) = 0.29289 \text{ (Cap height)} \\\)

\(\dfrac {1}{3} \pi * (0.29289)^2 * \left ( 3(1) -0.29289 \right ) = 0.24319 \text { (Cap volume)} \\ \)

Sv/Capv = ratio

4.1888/0.2432= 17.2237

Multiply this ratio by the 5 lbs.

5*17.223 = 86.12 lbs

One monster cheese ball, for sure! :)

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Sir CPhill’s method is the same, but it looks like he took a 1/4 of the radius instead of the circumference. 

Well Mr. Banker, if you’ve not fallen asleep on (or in) your china cabinet --or even if you have, have yourself a Merry little Christmas.   

GA