Expii Solve is calling… Because Solve set 52: Music is here!
This week’s Solve set is inspired by Music and the recent American Music Awards! Solve set 52 features all your favorite artists from Ariana Grande, Drake, Justin Bieber, The Chainsmokers and more! Practice math skills related to topics like periodic functions, Product-to-sum and sum-to-product identities and Average Value of a Function.
Take a look a Question 4: Electric Amp-litude
The Chainsmokers won the Electronic Dance Music Award at the 2016 AMA’s. The art of electronically mixing digital samples involves the addition of sound waves. When multiple sources are played simultaneously, the loudness of the resulting mix is quite nuanced. When two electronic synthesizers are played simultaneously with perfect sine waves of identical frequency and amplitude, it is theoretically possible for them to cancel each other out entirely, if each wave’s peak matches precisely with the other wave’s trough. On the other hand, when the waves are perfectly in phase (with peaks matching with peaks), the resulting amplitude is double that of each individual wave, with the same frequency. For all intermediate relative timing offsets between the waves, the resulting amplitude is somewhere in between, but the resulting frequency is the same as the original. In reality, each synthesizer’s sound comes from many frequencies overlaid upon each other, whose individual timing shifts are effectively randomly distributed relative to each other. Model this effect by considering the sum of two perfect sine waves with the same frequency and amplitude, but with peaks offset by a uniformly random time. What is the expected value of the ratio of the resulting squared amplitude, divided by the squared amplitude of one of the individual waves? Round your answer to the nearest thousandth.
Up for a challenge? Test out your Binomial probability skills with Questions 5: To Tweet, or Not to Tweet.
To vote for the 2016 American Music Awards, you could use their website or Twitter, and retweets of your vote count too. To understand the potential impact of retweets, consider a fictitious Twitterverse in which there are 1 billion accounts, each of which is followed by 200 random accounts. Suppose that when an account sends out a tweet, each of its followers independently has a 1% probability of retweeting that message, but the system is smart, and does not deliver the tweet to any account which already had seen it. What is the expected number of accounts that see a given tweet? Round your answer to the nearest million.