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Calculus Problem:
Harry Potter!

Welcome to the 1st calculus question of PCPOW!
Entries will be due by 6:00 pm PDT on Sunday, December 16th, 2001.
Good luck!

This year the calculus questions will be worth 9 points to help prepare all students taking AP Calculus for the AP Calculus Free Response section of the test. Have fun!!
Oh no! The Evil Wizard Voldemort has captured another unicorn. Can you help Harry get to the unicorn as fast as he can?

Harry's mail owl has informed him of the location of the trapped unicorn. The unicorn is trapped exactly 120 meters East of the edge of the forest (indicated with a red dot on the map below). Harry is currently 360 meters due West of the forest edge and is 500 meters from the unicorn (as the owl flies). His location is indicated with a purple dot on the map.

Harry leaps on his Nimbus 2000 Flying Broomstick and rushes to aid the unicorn.

  1. If Harry's speed in the courtyard is 30 meters per second and his speed in the woods is 8 meters per second (it is much harder to dodge all those trees), where should he enter the woods so that he reaches the unicorn as fast as possible? Justify your answer using calculus!
  2. Assuming Harry follows the fastest possible path (found in the last part of the question), how fast will it take him to get there?
  3. Voldemort is located on the same horizontal line as Harry but he is located 15 meters East of the forest edge. His position is indicated with a black dot. If he moves at 6 meters per second, will Harry beat him to the unicorn (assume Voldemort left at the same time Harry did).
  4. How fast does it take Voldemort to arrive at the unicorn?

Note: Map is not drawn to scale!

  See the solution!