Rich Math Problems

Plane Flying Overhead

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You are standing outside in the sunshine. An airplane at 3000 feet is going to fly directly over your head in a few minutes. It is traveling 200 mph. You don’t notice it until you hear it, and the instant you hear it you spot it at x degrees above the horizon. However, the sound you hear came from the plane somewhat earlier, when it was 20° above the horizon. Take the speed of sound at 1100 ft/sec, find x, the angle at which you looked up and saw the plane.

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Wind Turbine Height on Approach

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The top of an off-shore wind turbine can be sighted from the deck of your yacht from a point 252 meters from the base of the wind turbine with an angle of elevation that doubles as you sail 172 meters closer to the base of the wind turbine. How tall is the wind turbine?

  1. 144
  2. 150
  3. 12 √161
  4. 152
  5. 13 √143

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Latitude of Flight

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Take the circumference of the earth at the equator to be 24,000 miles. An airplane taking off at the equator and flying west at 1000 miles an hour would land at exactly the same time that it started (why?). Moreover, the sun would not move in the plane’s sky during the flight. (Work this out visually in your mind.) Now, at what degree of latitude could a plane flying 500 miles per hour keep up with the sun in this way?

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Plane Speed = f(Latitude)

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Take the circumference of the earth at the equator to be 24,000 miles. An airplane taking off at the equator and flying west at 1,000 miles per hour would land at exactly the same time that it started, and the sun would not move in the plane’s sky during that flight. (Work this out visually in your mind.) Now, can you find a formula so that the speed of a plane flying at any particular latitude and keeping up with the sun, is a function of that latitude: speed = f(x°)?

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Vertical Tower 1

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MA represents a vertical tower. Points A, B, and C lie in the same horizontal plane. MB and MC are wire stays, each of length 20 meters. The angle of elevation of M at B is x&deg. ∠MCB is z&deg, and ∠ABC = y&deg = ∠ACB.

  1. Prove that BC = 40 cos z.
  2. Show that BC = 40(cos x)(cos y).
  3. Calculate the magnitude of x if y = 42� and z = 71.8�.

Basketball Shot Contest

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A special basketball shot is made by James 20% of the time, by Kristy 30% of the time, and by Hannah 40% of the time. A game is played in which the players shoot consecutively and the first player to make the shot wins. If the order of shooting is first James, then Kristy, and then Hannah, which of the following statements are true?

  1. Kristy wins most often.
  2. Because Hannah is best at making this shot, she is more likely to win than
    Kristy.
  3. James will win least often.
  4. James, Kristy and Hannah all win the same number of games.
  5. James will win at least as often as either of the others.
  1. b, e
  2. d
  3. a, c
  4. b, c
  5. They are all false.

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