Sunday, May 14, 2006

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 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: 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.

Saturday, April 01, 2006

Episode Five A wicked Brother

A wicked BrotherPrince won the archey contest. But said Anna’s father Prospero, the Gonzalo of Calculand “Prince, 3 years ago,” continued prospero, “you were Prince of calculand, and also the duke of this calculand. Uou have a brother whose name was Antonio, to whom you trust everything, and you were fond of quiet and deep study, you commonly left the governing of your state affairs to your brother, your false brother Antonio being thus in possession of your power, began to think himself the duke indeed. The chance you gave him of making himself beloved of your subjects awakened in his bad nature a proud desire to rob you of your dukedom.” “He wanted to kill you, I think no wonder you lost your memory and found yourself at that cave.” “Why didn’t he just destroy me?” Asked Prince. “They dare not, so dear was the love that your people bore you.” Continued Prospero, Now Antonio is outside in Biologiland, and will return in two days, you need to catch him at the river Styx before his return

A) Find the area of the boat that Prince going to use. He will be have fight with Antonio on this boat.
Two functions about the boat, two sides of the boat. F(x)=4-x^2 and g(x)=5-2x.

B) The temperature on the river during 1 24-hour period is given by
F(t)= -( 80-100 cos (Pai/t)/12). 0Find the temperature between t=6 and t=14.

C) And storm will have whenever the temperature is at -78 degrees Fahrenheit, for what value of t is storm going to have?

Monday, March 27, 2006

The Archrival

Satiety. No. It was exhilaration. Never has he felt such pouring emotions as far as he remembered. Utterly gratifying every god he knew, fervently exalting the heavens up above, boy was he ecstatic. He, himself, could not elucidate what he just experienced, but surely he was certainly happy. He cared about nothing except for her.

"Are you alright? Did he harm you? Oh princess, I'm never keeping my eyes off of you!"

Silence. It was as if the angels passed by them. She smiled. He was bashful. It was evident. They have fallen in love with each other. Days went on and their affection grew stronger and stronger. But hence, Anna recalled, she cannot feel this. She must not commit to someone else when she is already about to marry Prince Rafael whom her father endorsed of.

"What am I doing? I am getting married. But I do not love Rafael, it is the Prince I want, the man who rescued me from agony. He is the one my heart is yearning for. He is the one I love. But father. Oh Father will be terrified! I do not want him to be upset. But I have no feelings for Rafael. What shall I do?"

Just as opposing thoughts were battling in her head, the King walked in anxiously with Prince Rafael.

"My daughter, thank heavens nothing happened to you. I'm here now. Upon hearing the incident, Prince Rafael immediately left his kingdom and went here to look out for you. Plus, I have great news. My child Anna and Prince Rafael of Biologiland, you will be joined in a holy matrimony as soon as we get started with the plan."

"But father. You know I do not love Rafael and I will not live the rest of my life regretting the day of my wedding. The man I love is the Prince of Calculand. My heart is only his."

"I will not possibly approve of him right away. I think it is just fair enough to know him a little better.Tomorrow there will be an archery contest. Whoever wins between Rafael and your prince will have your hand for marriage."

"But father!"

"No buts my little one. This is my decision. Let your man prove himself. "

How can the prince conceivably do this, there is no time for practice. And from what he heard, Prince Rafael used to be a champion at this event. He will not bear losing the girl he loves. But how? The round board they used was as little as 8 inches wide.

"Alas!" He said. "I am no good at aiming circles, but I know I am the best archer when it comes to quadrilaterals. (Funny we should ask. When he was a kid he had phobias of aiming at round things because the last time he did it, he hit the jester's right ear. Since then, he never tried doing archery on circles again) Well then I need to calculate the area of the rectangle inscribed in that circle to aim perfectly at its center."

a. What is the major equation required for this problem?
b. Find the critical values of x
c. What is the largest area the rectangle can have and what are the dimensions?

Tuesday, January 24, 2006

Episode 3: When the Prince met Anna

In the previous episode, the Prince met Charon, the Ferryman of the Underworld, who granted him three mathematical questions to answer before he was able to cross the river. After much anticipation, the Prince successfully answered all three questions and was allowed to continue across the river….
The Prince continued to travel on for many miles while never even realizing he had just entered the “Forbidden Forest”. Exhausted from his long journey, he sat under a large tree bearing delicious guava. Decided to eat and eventually fell asleep under the tree. Suddenly he was awake by a beautiful song, sung by a beautiful woman, followed by a wizard.
“I am Anna Kiki of Calculand and this is my sidekick Harrey the wizard”, she replies.
“Hello!” said the wizard.
“I…I am… I don’t know my name. I don’t remember who I am….”
“ Oh! I remember you. You are the Prince of…..” And before Anna say another word she was swept away instantaneously by a purple, green spotted fire-breathing dragon followed by a hairy short goblin.
“Who are you?” yells the Prince, drawing out his sword.
“I’m the great and powerful goblin of Dereva. Hemo the goblin. Ha!ha!ha!” said the evil goblin.
“Put her down this instant Hemo”
“Not so fast,” he says laughing in amusement, “If you want her save her then”.
“Barney! Take her to the castle” demanded Hemo to his pet dragon.
Then Barney flies Anna over to the goblins castle and places the princess in the tower leaving the prince and the wizard on the ground.
“Oh k! what do we do now?” asked the furious Prince.
“I have an idea!” said Harrey. With his magical powers he created a 72 foot ladder.
“Here, use this and lean this against the castle walls.”
“Thanks Harrey!”
Initially the prince lean the top of the ladder against the wall at a height of 45 feet. However, Anna is located on top of the castle wall at a height of approximately 56 feet. So Harrey use his magical powers and he push the base of the ladder towards the castle wall in order to increase its height. However, the evil goblin also uses his powers to stop the prince from reaching the top of the castle. Luckily, the force that Harrey pushed the ladder towards the castle wall was greater than the force of Hemo. Because of Harrey’s strength he was able to push the ladder on top of the wall. He knows that when the bottom of the ladder is 11 feet away from where it started, the Prince would reach the tower and save Anna. It took him 1 minute and 12 seconds to push the base of the ladder 11 feet. Then when the prince got to the top he drew out his sword and slayed Hemo‘s pet Barney. The goblin just vanished into thin air after the encounter. Anna kissed the Prince and whispered softly against his chest, “Thank you, your majesty!”

A.) How far away does the ladder initially away from the wall?
B.) At what rate I feet per second did Harrey the wizard pushed the ladder towards the wall?
C.) When the Prince reached the top of the castle at what rate was the top of the ladder moving?

Wednesday, December 21, 2005

Chapter Two: The Trip Across Styx

After getting past Cerebus, the prince found himself with in a vast cavern with a very odd river flowing through it. There was a dock with a cloaked man sitting crouched in his boat. Upon approaching, the man greeted the prince. "I am Charon, ferryman of the Underworld. You must pay the toll if you wish to cross the river Styx." The prince reached for his coin pouch but the ferryman stated, "Rules are different now. Hades decreed that I am now supposed to charge 1 gold coin for every 4 meters I take you. It usually takes me 29 seconds to cross the river. I manage the speed of my boat at 6 meters every second, but the blasted river makes my boat speed up at 4 meters per second every second." With no other way across the river of souls, the prince will have to pay the toll. Help the prince find his bill for the ferry ride.

A)What is the function that represents the velocity of the boat?

B) Given the time interval of the trip, how far will the boat travel before letting the prince off?

C)What will his final fare at the end of the trip be?