# Parametric/Polar EC

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• James Wand was a spy working under MI5: Magical Intelligence, Section 5. He had just been debriefed on an important mission, find and recover an ancient magical treasure from a large mansion belonging to a millionaire. His superiors did not know how exactly the house looked like, so James had to form a plan on how to get in and out.
• The Adventures of James Wand
• James Wand had just arrived at the millionaire's house, where his superiors believed was the important treasure to recover. He had done a lot of planning to scope out how he could infiltrate the house. He first went around to an unused service door that he had seen while looking through the blueprints for the house. There, he could slip in unnoticed.
• He first needed to determine the area of this strange keyhole in order to use his magical key on it. After comparing it with a coordinate plane, he determined that the keyhole was part of a cardioid from θ = 0 - pi/6 with the equation r=1+cosθ. With this, he was able to find that the area of the keyhole was 0.87 cm^2, and he transformed his magical key into the right area in order to open the door and enter the next area.
• Next, James Wand had to use his trusty laser cutter in order to cut through the painting in a precise circle with the parametric equations x=3cos(t), y= 3sin(t), with 0<t<2pi. In order to find the length he needed to cut, he had to calculate the arc length of the parametric curve. He cut a circle 6pi cm around a button within the painting that opened a door within the bookshelf and took him to the next area.
• James Wand reached a secret cave under the mansion. He noticed an opening with a with a keypad that had instructions saying "Slope: t=pi/4." He first noticed that the opening was shaped like an ellipse with the parametric equations: x=cos(t), y=3sin(t), 0<t<pi. With that information, he could find the slope of the line by taking the derivative. He found the slope to be -3 at t=pi/4, and he inputted the answer into the keypad to enter the next room.
• Finally, James Wand found the stolen treasure and teleported out of the creepy lair and back to MI5 headquarters. There, his superiors told him that he had just recovered an ancient wand that had the potential to destroy the world, and that he had saved humanity from evil people who would use that power for harm. What will James Wand have to do next? Find out next time in "The Adventures of James Wand"!