GingerAle

Sorry for the delay. Someone placed an order for a Pound Puppy.

Oh yes, Spicy salsa does even out the sweetness of Chihuahua. Lettuce is always a favorite.    

This is actually my second restaurant. My first one was called the Dog Food Cafe. Most thought it was a place they could bring their dog and order a snack for him from a menu. They were kind of put off by what was on the menu.    

My best and priciest menu items are Pekinese in Bark Sauce, BloodHound Sausage, Cream of Wheaton Terrier with Collie Flower, Weinerdog Schnitzel, Shih-tzu on a shingle, and French Poodle Fries,

For desserts they are Yorkshire Pudding Terrier,  Beagle with cream cheese, Chocolate and Butterscotch Lab Sundaes, and Great Danish.

Mr. Banker, I was playing monopoly with my dog and cat again, just waiting for you chime in.

My cat was the banker this time and he hadn’t embezzled any money yet.  I asked him why and he said he was too excited about reading your newest post. For some reason my cat likes you. There is no accounting for his taste though; he likes mice and rats–especially rats. 

Anyway, to the point: while the formula you present here is correct, there isn’t any explicit time involved in this question. Nope, just orders (or invoices, if you prefer). It might be a day, a week, a month, or years between the orders.  Your presentation would only be correct if the owner generated one sales invoice per year.  If that’s the case, he should fire himself. Probably, he just needs to fire his banker.

 I understand that in banker theory, time equals money, but really, it’s what one does in time that counts.   The asker can probably figure that out though, he posted a reasonably articulated question, so he probably knows to substitute the order number for the time in this equation—which is impressive considering a banker didn’t.

You know, there’s a human who works for Lancelot Link’s A.P.E Company; he’s one of the very few humans in his employ. He’s the company’s principal investment banker.  Now, I would have thought that a genetically enhanced chimp would fill such an important position. I asked Lancelot why. He explained that while a genetically enhanced chimp would be vastly superior to a human, it would also be difficult to detect the sophisticated embezzlement schemes he would capitalize on. It’s much, much easier to catch humans. 

Chimp Loki discovered a genetic marker that is very common in human bankers and extremely common in investment bankers. I asked chimp Loki to explain. What he said was quite funny, but shocking–so shocking, I’m sure one or more of the moderators will delete it, after reading it –maybe while laughing.  Because of the deletion risk, I will include it in a separate post. 

Solution(s)

\(\text {(a) }\\ \text {Find coordinates (y) for masses } \displaystyle m_1(y_1) \text { and } m_2(y_2)\\ y_1 = 0.5(9.81)*(0.300)^2 = 0.441 \small \text{meters}\\ y_2 = 0.5(9.81)*(0.300 – 0.100)^2 = 0.196 \small \text{ meters}\\ \\ \small \text{ Formula for the center of mass (2 objects):}\\ \\ y_{com} = \dfrac{(m_1y_1 + m_2y_2)}{(m_1 + m_2)}\\ y_{com} = \dfrac{((1) (0.441) + (2)(0.196)}{(1) + (2)} = 0.278 \small \text{ meters} \leftarrow \small\color{green} \text{Answer for (a)}\\ \text {(b) }\\ \text{Velocity of center of mass}\\ v1=(g)(t)_1 = 9.8*0.300 = \small \text{ 2.94m/s }\\ v2=(g)(t)_2 = 9.8*0.200 = \small \text{ 1.96m /s }\\ v_{com} = \dfrac{((1)(2.94)+(2)(1.96))}{3} = 2.287 \small \text{ m/s} \leftarrow \small\color{green} \text{Answer for (b)}\\\\ \small \text{ }\\ \small \text{Theory & Formulas: Complements of Archimedes of Syracuse, Leonhard Euler, Sir Isaac Newton, Pappus of Alexandria,}\\ \small \text{Guido Ubaldi, Francesco Maurolico, Federico Commandino, Simon Stevin, Luca Valerio, Jean-Charles de la Faille, Paul Guldin,}\\ \small \text{John Wallis, Louis Carré, Pierre Varignon, and Alexis Clairaut.}\\ \small \text{ } \hspace{20em} \scriptsize \text {(Probably a few others, too. The COM concept is ancient) }\\ \small \text{Produced by Lancelot Link and company. }\\ \small \text{Directed by GingerAle. }\\ \small \text{Sponsored by Naus Corp. Quantum Pharmaceuticals. }\\ \small \text{ } \hspace{15em} \scriptsize \text { Making a better world by neutralizing the quantum dumbness of }\\ \small \text{ } \hspace{15em} \scriptsize \text { Blarney Masters and related dumb-dumbs }\\\)

Hi Max,

There’s an omission in your equation.  The Sigma needs squared.

\(\large \sigma^2 =\normalsize \dfrac{1}{N-1}\left(\displaystyle\sum^{N-1}_{i=0}x^{2}_{i}-\dfrac{1}{N}\left(\displaystyle\sum^{N-1}_{i=0}x_i\right)^2\right)\)

Or use the square root of the equation:

\(\large \sigma =\normalsize \sqrt{\dfrac{1}{N-1}\left(\displaystyle\sum^{N-1}_{i=0}x^{2}_{i}-\dfrac{1}{N}\left(\displaystyle\sum^{N-1}_{i=0}x_i\right)^2\right)}\)

This is an efficient computational form of the equivalent:

\(\sigma = \sqrt { \frac{1}{N} \sum \limits_{i=1} ^ {N}(\chi - \mu)^2 }\)

--------------

The formula for circular standard deviation is in post #3

Directional statistics (also circular statistics or spherical statistics) is the subdiscipline of statistics that deals with directions (unit vectors in Rn), axes (lines through the origin in Rn) or rotations in Rn. More generally, directional statistics deals with observations on compact Riemannian manifolds. (https://en.wikipedia.org/wiki/Directional_statistics)

Here are the equations for Circular Standard Deviation (from the same source noted above):

\(S(z)={\sqrt {\ln(1/R^{2})}}={\sqrt {-2\ln(R)}}\small \hspace{25pt} \text {Equation for Sample Circular Standard Deviation } \\ {\overline {S}}(z)={\sqrt {\ln(1/{\overline {R}}^{2})}}={\sqrt {-2\ln({\overline {R}})}} \hspace{24pt} \small \text {Equation for Population Circular Standard Deviation } \\\)