Wednesday, August 22, 2012

Winners' Circle: Math and Word Problems Solved!

A mere two days ago, our Contest of the Week was announced, presenting the complex challenge of identifying a single word and seven digit sequence, out of infinitely many possible combinations, often spoken by our lead singer and echoed by the rest of us. Within seconds, winning entries were piling up, giving us an unprecedented tie, three runners up, and a legit alternative correct answer!

Donita Hillman-Jacobsen, through her proxy Jamie, was tied for first place with the target answers 's-a-f-e-t-y' and 8675309, and had these brainteasers for us:

"How many drums does it take Josh of Seagulls to play a gig?"

Brendan Goes to Hollywood: Including the secret drums positioned behind the stage and played by Josh's hologram throughout the show, plus the spare snares he has stashed away, approximately 52.3214 drums are needed.

"How many glasses of water did it take Alex in Chains to get through the Fat City gig?"

Alex in Chains: According to the Daily Hydration Calculator, I require 85.5 ounces (2.6 liters) of water per day. With a healthy diet, about 20 percent of water is from food, leaving the need for an additional 68.4 ounces of water, or 2.1 liters, daily. I used exactly 4.2 liters of water, half to stay hydrated and half to keep Punky Drewster's head and shirt drenched throughout the evening.


Andrea Brenner Greaney, also tied for first place, and Anna Summers and Caron Barnhart, runners up, did not need our assistance with any word or math problem.

Josh of Seagulls: I only count to four.


Jesse Van Halen: The word IS the problem.


Uncle Jeff, Astrophysicist and winner with a surprise alternative answer ('r-o-c-k' in the USA, 8675309), challenged the band with:

"What do you get when you integrate 4/(1+t^2) from t=0 to t=1?"

Punky Drewster: One way to find the answer is to make a right triangle with legs of length 1 and t so the hypotenuse has length sqrt(1+t^2). In this triangle, define theta as the angle opposite the side of length t. This lets sin(theta)=t/sqrt(1+t^2), cos(theta)=1/sqrt(1+t^2), [cos(theta)]^2=1/(1+t^2), and tan(theta)=t. With these relationships, you can show that dt=d(theta)/[cos(theta)]^2 and theta=arctan(t). Then you can change the integral to the integral of 4 from theta=0 to theta=pi/4.

Lynndi Lauper: Another approach is to use complex numbers (ones that contain i, where i=sqrt(-1) and i^2 = -1). You can write 4/(1+t^2)=4/[(1 + i t)(1 - i t)]=2/(1 + i t) + 2/(1 - i t). Then, the integral of 4/(1+t^2) from t=0 to t=1 becomes (2/i)ln[(1+i)/(1-i)] which equals (2/i)ln(i). Then, since i^2 = -1 = e^(i pi), we have i=sqrt(-1)=e^(i pi/2). This gives ln(i)=ln[e^(i pi/2)]=(i pi/2). ln(x) means the natural logarithm of x. The natural logarithm has base e. This lets ln(e^x)=[x ln(e)]=x hold. e^x means e (around 2.718) raised to the x power. ln(e) equals 1. Putting all the pieces together, you end with the integral of 4/(1+t^2) from t=0 to t=1 being equal to pi.


Monday, August 20, 2012

Contest of the Week: Spelling Bee & When is Math Ever a Problem?

For sixteen years, we practiced spelling just one word. Most of us have it now, though some simply repeat whatever the lead singer says. We also memorized a seven digit sequence, with similar result. (Hint: the number is neither pi nor 69).

If you are the first to guess the word and the number we've mastered, you can win a beautiful plastic trophy and our assistance with a word or math problem!