Sunday, April 5, 2009

The thrilling conclusion to "Do you remember SAT math?"

Two weeks ago, I posted two SAT-style Quantitative Comparison questions. This week, I am posting the solutions (sorry, I know it's a bit late).

If we let the radius of the inner circle be r and the length of segment BC be s such that the radius of the outer circle is r+s, then we can express all the quantities that we are interested in in terms of r and s.

The length of arc CE is:
¼(2π(r+s)) = ½πr + ½πs

The length of arc BD plus the length of segment DE is:
¼(2πr) + s = ½πr + s

The length of segment CB plus arc BD plus the length of segment DE is:
s +¼(2πr) + s = ½πr + 2s

When comparing each pair of terms, the ½πr cancel each other out. Because s is a length, we know that s>0, which means that ½πs is always greater than s, but is always less than 2s. In terms of answering the quantitative comparison questions, this means the answer to the first question must be A and the answer to the second question must be B.

Put another way, if you are walking around a 90 degree arc (such as from point C to point E), it will always be faster to walk around the outer arc than to walk to an inner arc, traverse it, and return to the outer arc. What is particularly interesting is that this is true regardless of the radius of the inner arc.

Now before you start taking the outside edge of every curve, note that this only holds true for arcs less than a certain size. For example, what if it you were traversing a semicircle instead?

The length of the outer arc would be:
½(2π(r+s)) = πrs

The length of the two straightaways plus the inner arc would be:
s +½(2πr) + s = πr + 2s

Because πs is greater than 2s, it makes more sense to walk to the inner arc when walking around a semicircle! So where is the breaking point? This can be determined by letting x be the fraction of the circle to traverse and setting the two path lengths equal to one another and solving for x:

x(2π(r+s)) = x(2πr) + 2s
xr+2πxs = 2πxr + 2s
xs = 2s
πx = 1
x = 1 / π

Plugging that into Google Calculator, we see that x is ~0.318, so once you're walking more than a third of the way around a circle, it makes more sense to move to the inner arc.

I thought that this would make a great SAT question for several reasons. First, and most importantly, this is a quantitative comparison question that does not have any numbers, yet its answer is not D! I'm pretty sure that most students hated geometry (so sad!), so when they come to something like this, it's pretty easy to throw up one's hands and declare the question unsolvable. Of course such a student would be incorrect...

Also, this question is resistant to some common SAT-solving techniques. For example, for geometry questions where the figure is to scale (such as this one), The Princeton Review will tell you to make small marks on your answer sheet so you can use it as a ruler, but because this problem involves arcs, that is of little help. Another common SAT tip is to redraw the diagram, exaggerating r or s, but if you play around with that, you can probably convince yourself that there are cases where one quantity is larger than the other, but it's hard to say conclusively that it holds true for arbitrary values of r and s.

That means the only ones who will be answering the question correctly are those who can do the math. Isn't that how it should be?

40 comments:

  1. The ultimate goal of online psychology research paper writing services is to provide Psychology Assignment Writing Services and psychology research paper services since most psychology coursework writing service students lack time to complete their custom psychology coursework writing services.

    ReplyDelete
  2. Students find Human Resource Writing Services as being of great assistance since they are able to complete their human resource assignment writing services and human resource research paper writing services on time.

    ReplyDelete
  3. This is interesting article..Thanks for sharing this wonderful information. 온라인카지노

    ReplyDelete
  4. Ahaa, its good discussion about this post here at this webpage, I have read all that, so at this time me also commenting
    here. 온라인카지노

    ReplyDelete
  5. 카지노사이트 What's up it's me, I am also visiting this website daily, this website
    is genuinely nice and the visitors are in fact sharing fastidious thoughts.

    ReplyDelete
  6. 토토사이트 Useful info. Lucky me I found your web site
    by chance, and I am surprised why this twist of fate
    didn't came about in advance! I bookmarked it.

    ReplyDelete
  7. 스포츠토토 I'm extremely impressed with your writing skills as well as with the
    layout on your blog. Is this a paid theme or did you modify it yourself?

    Either way keep up the nice quality writing, it's rare
    to see a great blog like this one these days.


    ReplyDelete
  8. Thanks for the blog loaded with so many information. Stopping by your blog helped me to get what I was looking for.
    섯다

    ReplyDelete
  9. I like your post. It is good to see you verbalize from the heart and clarity on this important subject can be easily observed..
    스포츠토토

    ReplyDelete
  10. This is an excellent post I seen thanks to share it. It is really what I wanted to see hope in future you will continue for sharing such a excellent post.
    일본야동

    ReplyDelete
  11. Уour blog providеd us useful information to work on. Үou have done a marvelous job! 바카라사이트

    ReplyDelete
  12. Your post has really helped me a lot. I live in a different country than you, but I believe it will help a lot in my country. 카지노사이트 Thank you very much. Can I refer to your post on my website? Your post touched me a lot and helped me a lot. If you have any questions, please visit my site and read what kind of posts I am posting. I am sure it will be interesting.


    ReplyDelete
  13. I am someone who works on the web. Sometimes I like to visit overseas sites sòng bạc when I have time. Among them, I like the site related to , but your site seems to be optimized for too!! Very good

    ReplyDelete
  14. Hey what a brilliant post I have come across and believe me I have been searching out for this similar kind of post for past a week and hardly came across this. Thank you very much and will look for more postings from you. 먹튀검증 and I am very happy to see your post just in time and it was a great help. Thank you ! Leave your blog address below. Please visit me anytime!

    ReplyDelete
  15. When I read your article on this topic, the first thought seems profound and difficult. There is also a bulletin board for discussion of articles and photos similar to this topic on my site, but I would like to visit once when I have time to discuss this topic. sòng bạc


    ReplyDelete
  16. I visited last Monday, and in the meantime, I came back in baccarat anticipation that there might be other articles related to I know there is no regret and leave a comment. Your related articles are very good, keep going!!

    ReplyDelete
  17. It's really great. Thank you for providing a quality article. There is something you might be interested in. Do you know 룰렛사이트? If you have more questions, please come to my site and check it out!


    ReplyDelete
  18. This is the post I was looking for 메이저사이트 I am very happy to finally read about the Thank you very much. Your post was of great help to me. If you are interested in the column I wrote, please visit my site .

    ReplyDelete
  19. It is included in my habit that I often visit blogs in my free time, so after landing on your blog. I have thoroughly impressed with it and decided to take out some precious time to visit it again and again. Thanks.สล็อตแตกง่าย

    ReplyDelete
  20. Are you the one who studies this subject?? I have a headache with this subject. 우리카지노 Looking at your writing was very helpful.

    ReplyDelete
  21. Hello. I'm subscribed to your posts. You inspire me a lot. 바카라사이트 I am so grateful.

    ReplyDelete
  22. We are looking for a lot of data on this item. In the meantime, this is the perfect article I was looking for . Please post a lot about items related to 메이저사이트추천 !!! I am waiting for your article. And when you are having difficulty writing articles, I think you can get a lot of help by visiting my .

    ReplyDelete
  23. Caesars Casino and Racetrack – 2021 New Jersey Gambling
    Caesars septcasino Resort aprcasino Casino & Racetrack is the latest casino worrione.com in New worrione Jersey to 도레미시디 출장샵 undergo a comprehensive safety review. The casino is owned by Caesars

    ReplyDelete
  24. pg slot เกมยอดนิยม เกมยอดฮิต pg slot ค่ายเกมส์ สล็อต น้องใหม่ที่มาแรง ที่พร้อมเปิดให้บริการท่านในต้นปี 63 ที่จะถึงนี้ มาพร้อมกับเกมส์ สล็อต รูปแบบใหม่ ที่น่าตื่นตาตื่นใจ

    ReplyDelete
  25. เล่นง่ายได้เงินจริง ท่องเที่ยวไปยังโลกของเงินตราด้วยกันที่เว็บไซต์ SLOT PG เว็บไซต์ที่มาพร้อมกับเกมออนไลน์หลากหลายรูปแบบ มีทั้งเกมแนวการผจญภัย แนวเกมแฟนตาซี แนวเกมการต่อสู้สุดเร้าใจ ตื่นตาตื่นใจไปกับภาพกราฟิกที่ถูกสร้างมาอย่างดีไม่ซ้ำใคร มีภาพที่คมชัดระดับ FULL HD มาพร้อมกับเงินรางวัลและฟีเจอร์แสนสนุกแจกให้ตลอดทั้งเกม

    ReplyDelete
  26. เว็บเกมของเรามีการอัพเดทเกมตลอดเวลา แจ็คพอตแตกง่าย โบนัสแตกไว จ่ายเงินจริง สมัครสมาชิกและรับสะดวกสบายไปกับระบบออโต้สุดลื่นไหลใน SLOTPG เว็บเกมสล็อตสุดทันสมัย ที่มาพร้อมรูปแบบใหม่สุดล้ำสมัยและมากที่สุดในปี 2022 ตัวเกมของเราโดดเด่นในเรื่องของระบบเกมและภาพกราฟิกในเกม รับรองว่าระบบเกมของเราช่วยสร้างความสนุกสนานและตื่นเต้นเร้าใจให้ผู้เล่นได้ทุกนาทีอย่างต่อเนื่อง หากท่านใดสนใจที่จะสมัครสมาชิกกับเราวันนี้ คุณจะได้รับโบนัสและแจ็คพอตใหญ่ในเกมไปเลยทันที 100% !!

    ReplyDelete
  27. Please let me know if you’re looking for a article writer for your site. You have some really great posts and I feel I would be a good asset. If you ever want to take some of the load off, I’d absolutely love to write some material for your blog in exchange for a link back to mine. Please send me an email if interested. Thank you 먹튀사이트 I couldn't think of this, but it's amazing! I wrote several posts similar to this one, but please come and see!!

    ReplyDelete
  28. เว็บแจกเครดิตฟรี 200 แค่สมัคร ไม่ต้องลงทุนเยอะ สมัครฟรีกับ ufac4 สมัครฟรี เล่นพนันออนไลน์ เว็บตรงไม่มีขั้นต่ำ สล็อตรองรับ วอเลท

    ReplyDelete
  29. I pay to read some web pages and site articles and reviews every day, based on features other than this web page.

    스포츠토토
    안전놀이터
    우리카지노탑
    카지노사이트

    ReplyDelete
  30. เมก้าสล็อต บริการเกมสล็อตออนไลน์มากกว่า 1,000 เกม สมัครสมาชิกรับโบนัส 100% ฝาก-ถอน ไม่มีขั้นต่ำ ใช้เวลาเพียง 10 วินาที รองรับระบบทรูมันนี่วอเลท True Wallet https://www.megagame.ltd

    ReplyDelete
  31. และที่สำคัญเว็บไซต์ของเรายังมีระบบฝากถอนออนไลน์แบบ pgslot สามารถถอนเงินได้ง่าย ๆ เงินเข้าไวภายใน 1 นาทีเท่านั้น แถมยังสามารถเล่นเกม ได้ง่าย ๆ ตลอดทุกที่ทุกเวลาบนสมาร์ตโฟนทั้งในระบบ IOS และ Android เป็นเว็บไซต์ที่สร้างมาเพื่อความสนุกสนานจากการเล่นเกมเดิมพันออนไลน์โดยเฉพาะ

    ReplyDelete
  32. การให้บริการของทางคาสิโนออนไลน์นั้นอยู่ในรูปแบบเว็บไซต์ และแอปพลิเคชั่น ที่ผู้เล่นสามรถเลือกได้ตามความสะดวกและการรองรับของอุปกรณ์ที่ใช้งานได้เลย และการเชื่อมต่อผ่านอินเทอร์เน็ตที่ทำให้สามารถเลนได้ทั่วทุกพื้นที่ไม่ว่าจะอยู่ที่ไหนก็ตาม เรียกได้ครบวงจรความบันเทิง
    สล็อต PG

    ReplyDelete
  33. เว็บหวยออนไลน์ที่ให้ราคาดีที่สุด !ทั้งหวยไทย หวยเวียดนาม หวยลาว ไม่มีกั๊ก ไม่มีอั้น ได้เลยเต็มๆ

    ReplyDelete
  34. Thank you for giving me this amazing information. You can also find more here 안전놀이터추천

    ReplyDelete
  35. เล่นได้เท่าไหร่ก็สามารถถอนได้ไม่จำกัดครั้ง ไม่ต้องโยกเงิน กระเป๋าเดียว ยืนยันอีกเสียงว่าแจกจริง!! พร้อมแตก แจกหนัก รวมค่ายยอดนิยมเว็บเดียวจบไม่ต้องเปลี่ยน ชวนเพื่อนรับค่าคอมมิชชั่น 10 % แจกหนักกว่าทุกเว็บ สล็อตjoker

    ReplyDelete