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?

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