What an annoying weekend I had. It’s all my own poor planning, but the Math Guide went temporarily out of stock this weekend. Hopefully, everything will be back to normal by tomorrow, which means I can get right back to giving books away!
This weekend challenge is about those crazy pirate ship rides they have at amusement parks. If you’ve never seen one of these in action, here’s a video. They basically swing you around in a big circle.
Since there are so many ways to win a Math Guide these days, I’m not gonna go easy on you guys with the challenge questions. Just remember: Challenge questions are not SAT questions, they’re just for me to have fun writing and you to suffer through solving. 🙂
Anyway, here we go.
Note: Figure not drawn to scale.
Larry is seated at point L on a Pirate Ship ride that has not started moving yet, as shown. The measure of ∠ABC is 60º, and the distance between points A and C is 35 feet. With the ride at rest, the lowest point of the pirate ship is 5 feet above the ground, and Larry is 6 feet off the ground. Once the ride is in motion, how long is the arc Larry travels between the two points at which he is 40 feet off the ground? (Assume that the ship is a 60º arc.)
Put your answers in the comments. First correct (non-anonymous) answer gets a Math Guide!
UPDATE: wgreens got it! Nice work.
Solution below the cut
The first (and probably most important thing) to recognize is that AB and BC are both radii of the circle on which the ship travels, which means △ABC is isosceles and angles A and C are congruent. Since we also know that the measure of ∠ABC is 60º, it turns out the triangle is equilateral, and thus the circle has a radius of 35. Boom.
This, as wgreens pointed out in his answer, has some convenient implications. Specifically, it means that the center of the circle is 40 feet off the ground, so the two points at which Larry is 40 feet off the ground lie on a diameter of the circle.
That means to find the arc length, we don’t need to do any more work with angles. We just need to find the circumference, and cut it in half.
C = 2πr
C = 2π(35)
C = 70π
Since Larry only travels half the length of the circumference, he travels 35π feet.