Episode 6: Darth Euler and the Phoenix Flame
To defeat Darth Euler, the sword Prince possesses will not be enough, a rare element exists where the Phoenix flame can be found, there the sword can be reforged and Prince will be able to defeat Darth Euler. The Phoenix flame exists near Atlantia which is convenient for Prince. As Prince arrives, he finds the element which is in crystal form, the problem with enhancing the sword, is the flame is too hot, if the crystal is heated to a certain temperature it becomes dangerously reactive, so reactive that if that temperature is reached or exceeded, the air around Prince and the element will become toxic. The Phoenix flame's temperature varies with the height of the flame, at the top, it is at it's coolest, at the bottom it is at it's hottest.
The rate at which the temperature changes due to the change in height can be represented by the differential equation: dT/dh = k(T -5050). When the height is 0 cm, the temperature is 5000 Kelvin. At the top of the flame at 150 cm above the ground, the temperature is coolest at 3610.5405 Kelvin or K. Prince must solve for the function that relates the temperature to the height and use that to find the exact height at which the temperature is 4444 Kelvin.
a) Find the general solution for the differential equation.
b) Find the exact solution for the differential equation using the form of (h,T) for the points (0,5000) and (150,3610.5405).
c) Help Prince solve for the exact height that the element needs to be held at in order for Prince to use the element as an upgrade for the sword.
Once the sword has been enhanced Darth Euler, can be removed from his position as dictator of Atlantia. To remove Darth Euler, Prince has to beat him at sword play, and Darth Euler happens to be a Master Swordsman. The element is to Darth Euler as Kryptonite is to Superman. Prince of Calculand will retain his rightful title as Duke of Calculand with his newly married wife Anna.