### Episode 6: Darth Euler and the Phoenix Flame

After solving the problem that Antonio had created, Prince of Calculand was starting from where he left off 3 years ago. Unfortunately for Prince, he could not foresee the emergence of an individual who has gained great power during Prince's absence. Calculand is vast, but at it's center is the great city of Atlantia where the ruler is an individual by the name of

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:

a) Find the general solution for the differential equation.

b) Find the exact solution for the differential equation using the form of

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.

*Darth Euler*. Darth Euler's face has never been seen by any living being. Darth Euler is not a normal person, he is a creature that happens to look like a Calculandian. He has produced a means that has allowed him to produce a large army that is only loyal to him and no other.He has achieved this by using what he calls,*Euler's Method*. In Atlantia, if Darth Euler is not embraced as sole ruler and god, he has them executed. The city of Atlantia is home to over 90% of Calculand's residents. The city of Atlantia is no longer a safe and friendly home, but rather the opposite since Darth Euler's rise of power. Atlantia is so special, because the people of Calculand are so intelligent that when they built the city, it was capable of hovering above the ground. The discontinuity caused by Darth Euler has allowed him to stay in power, because this discontinuity is preventing the greatest minds in Calculand from finding a solution that will defeat him and his harsh rule over the citizens of Atlantia.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:

**When the height is***dT/dh = k(T -5050).**, the temperature is***0 cm***. At the top of the flame at***5000 Kelvin***above the ground, the temperature is coolest at***150 cm***or***3610.5405 Kelvin***. 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***K***.***4444 Kelvin**a) Find the general solution for the differential equation.

b) Find the exact solution for the differential equation using the form of

*for the points***(h,T)***and***(0,5000)***.***(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.