Monday, April 29, 2013

Precal Problems

I'm having fun writing some problems for my precal exam and review, so I thought I'd share some of what I've got so far.  Feel free to use.  Where necessary, numbers were researched and are accurate:

1. (Law of Sines/Cosines) You Quantum Leap into the body of an artist who works on building furniture from reclaimed wood.  Ziggy and Al agree that you have a 98% change of moving on if you create a triangular table top from the three pieces of wood that remain from your wife's crib.  The pieces measure 3 feet, 2.5 feet, and 4 feet.  Without cutting any of the pieces, at what angles should you attach the ends of the pieces to one another to create this amazing table?  (Find all three angles and pray that this one will be the leap home!)

2. (Right Triangle Trig) You are marching your army towards "The Wall" away from the White Walkers towards the 7 Kingdoms.  You know this wall is 700 feet tall, but need to figure out how far away you are to know how much food you need to take from Craster's stores.  One of your scouts runs ahead 4 miles and from there sights the top of the wall at 1º from his position.  How far is your camp from the Wall? (Answer in miles. 5280 feet = 1 mile).

3.  (Right Triangle Trig)  It's 1895 and you are emigrating to the United States!  To prove your worth to your new country so they will let you in, you decide to do some quick calculations while on the boat ride to Ellis Island.  From some distance away, you initially sight the top of the Statue of Liberty at an angle of elevation of 3º.  Based on the ship's speed you calculate that you travel 1460 feet closer right when the angle of elevation becomes 4º.  To the nearest foot, how tall is the Statue of Liberty?  Your future citizenship may depend on it!

4. (Angular Motion) The Curiosity Mars rover has a weird pattern on its wheels that includes the morse code for ``JPL'' (Jet Propulsion Lab, the arm of NASA driving the rover).  This helps the scientists figure out how far the rover has traveled by seeing the pattern in the tracks left behind while driving.  The wheels have a 50 cm diameter.  How many copies of JPL are left in the ground when the rover has traveled 5km?  If it takes the rover 3 days to travel this distance, what is its average speed in kilometers per hour?

5. (Angular Motion) A "45" was a vinyl record with a single on each side.  It was called that because it would spin at 45 revolutions per minute.  The standard size is a 7 inch diameter.
(a.) A fat ant wants to slim down his exoskeleton by running along the edge of the outer edge of the record while it plays the entirety of The Monkees' ``I'm a Believer'' (2 minutes 47 seconds long).  How far will it have run by the end of the song?
(b.) The ant's friend wants to keep her company, but doesn't really need as rigorous of a workout.  So, she's going to run around the hole which only has a 1.5 inch diameter.  By the end of the same song, how far will this ant have run?

6. (Angular Motion) Your car came with 17 inch diameter wheels.
(a.) How many times will a wheel rotate in one mile? (1 mile = 5280 feet.  1 foot = 12 inches.)
(b.) You want to pimp your ride up to 22 inch rims.  How far will you have actually gone in the new rims when they rotate the same amount as the answer from part (a)? (Answer in miles.)
(c.) If you don't recalibrate the sensors in your car, your spedometer may say you are going 60 mph on the interstate, but how fast are you actually going in the new wheels? (Does your answer make sense?  Should it be faster or slower than the car says? This has happened to me!)

7. (Right Triangle Trig) After the Mars Curiosity rover landed, it needed to assess where it was in relation to one of its goals, Aeolis Mons (aka Mount Sharp).  NASA knows that the mountain is supposed to be 18,000 feet tall (about the same size as Mount McKinley, the tallest mountain in North America).  Initial pictures show that the angle to the top of the mountain from Curiosity's current position was 18.71º.  What is the horizontal distance from the rover to the mountain (round to the tens place)?

8. (Law of Sines/Cosines) You wake up in a daze in the middle of a forest.  You've been kidnapped!  Fortunately, your captors left your school bag with you and (oddly) a map.  From the map you can tell that if you walk at a bearing of 53º, you'll find your way to civilization.  In your bag you also have a ruler and calculator.  You find two straight sticks and measure them to be 8 inches and 5 inches.  If you put two ends of the sticks together, you should be able to form a 53º angle and find your way.  How far apart (in inches to three decimal places) should the other ends of the sticks be to create a 53º?