# Project #62640 - Mathematic Modeling

This is a project that has to be done in MapleSoft

1. a)Ã‚Â  Animate the full rotation of the parametric curve P: x = 5 cos(t) + 7, y = 13 sin(t) + 21, 0 <= t and t <= 2 Pi about the center point (-8, -15) leaving 10 frames behind.

b) Animate the full rotation of the parametric curve P:
Ã‚Â Ã‚Â Ã‚Â Ã‚Â Ã‚Â  x = 4 cos(3 t), y = 4 sin(6 t), 0 <= t and t <= 2 Pi
Ã‚Â about the center point (-10, -15) leaving 5 frames behind

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Ã‚Â Ã‚Â Ã‚Â Ã‚Â  2.Ã‚Â Ã‚Â  a) Plot the spacecurve x = 5 cos(2 t) - 3 sin(t), y = 2 sin(2 t) + cos(t), z = 5 t + 2 from t = 0 to t = 4 Pi with axes boxed.

b) Plot the parametric surface x = 5 cos(u) sin(v), y = 5 cos(u) cos(v), z = 8 sin(u)Ã‚Â  for 0 <= u and u <= 2 Pi and 0 <= v and v <= 2 Pi with axes boxed.

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Ã‚Â Ã‚Â Ã‚Â Ã‚Â Ã‚Â  3.Ã‚Â  a) Use the N-R algorithm to find the value of
Ã‚Â Ã‚Â Ã‚Â Ã‚Â Ã‚Â Ã‚Â Ã‚Â Ã‚Â Ã‚Â Ã‚Â Ã‚Â Ã‚Â Ã‚Â Ã‚Â Ã‚Â Ã‚Â Ã‚Â Ã‚Â Ã‚Â Ã‚Â Ã‚Â Ã‚Â Ã‚Â Ã‚Â Ã‚Â Ã‚Â Ã‚Â Ã‚Â Ã‚Â Ã‚Â  (1/7)

Ã‚Â  Ã‚Â  Ã‚Â  Ã‚Â  Ã‚Â  Ã‚Â  Ã‚Â  Ã‚Â  Ã‚Â  Ã‚Â  Ã‚Â  Ã‚Â  Ã‚Â  Ã‚Â  Pi^Ã‚Â  Ã‚Â Ã‚Â Ã‚Â Ã‚Â Ã‚Â Ã‚Â Ã‚Â  accurate to 10 decimal places.Ã‚Â  Use an integer starting guess.

b) Use the system version of the N-R algorithm to find the first quadrant intersection point of the two curves
Ã‚Â Ã‚Â Ã‚Â Ã‚Â Ã‚Â Ã‚Â Ã‚Â Ã‚Â Ã‚Â Ã‚Â Ã‚Â Ã‚Â Ã‚Â Ã‚Â Ã‚Â Ã‚Â Ã‚Â Ã‚Â Ã‚Â Ã‚Â Ã‚Â Ã‚Â Ã‚Â Ã‚Â Ã‚Â Ã‚Â  2Ã‚Â Ã‚Â Ã‚Â  2Ã‚Â Ã‚Â Ã‚Â Ã‚Â
Ã‚Â Ã‚Â Ã‚Â Ã‚Â Ã‚Â Ã‚Â Ã‚Â Ã‚Â Ã‚Â Ã‚Â Ã‚Â Ã‚Â Ã‚Â Ã‚Â Ã‚Â Ã‚Â Ã‚Â Ã‚Â Ã‚Â Ã‚Â Ã‚Â Ã‚Â Ã‚Â Ã‚Â Ã‚Â  xÃ‚Â  + yÃ‚Â  = 10
Ã‚Â and
Ã‚Â Ã‚Â Ã‚Â Ã‚Â Ã‚Â Ã‚Â Ã‚Â Ã‚Â Ã‚Â Ã‚Â Ã‚Â Ã‚Â Ã‚Â Ã‚Â Ã‚Â Ã‚Â Ã‚Â Ã‚Â Ã‚Â Ã‚Â Ã‚Â Ã‚Â Ã‚Â Ã‚Â Ã‚Â Ã‚Â  2Ã‚Â Ã‚Â Ã‚Â  2Ã‚Â Ã‚Â Ã‚Â
Ã‚Â Ã‚Â Ã‚Â Ã‚Â Ã‚Â Ã‚Â Ã‚Â Ã‚Â Ã‚Â Ã‚Â Ã‚Â Ã‚Â Ã‚Â Ã‚Â Ã‚Â Ã‚Â Ã‚Â Ã‚Â Ã‚Â Ã‚Â Ã‚Â Ã‚Â Ã‚Â Ã‚Â Ã‚Â  xÃ‚Â  - yÃ‚Â  = 1
.Ã‚Â  Use an integer starting guess and find the answer accurate to 10 decimal places.

Ã‚Â Ã‚Â Ã‚Â Ã‚Â Ã‚Â Ã‚Â Ã‚Â Ã‚Â  4.Ã‚Â  a) Find and plot the trajectory of the projectile launched from a height H = 2500 feet above groundlevel towards a target at groundlevel 9000 feet from the launcher.Ã‚Â  The projectile must hit the target with the angle of elevation which gives the smallest launch speed needed to hit the target.Ã‚Â  Assume gravity and air resistance are the two forces acting on the projectile after launch with a coefficient of friction of K = 0.025.

b) Find the time T in seconds at which the projectile hits the target.

Ã‚Â Ã‚Â Ã‚Â Ã‚Â Ã‚Â Ã‚Â Ã‚Â Ã‚Â  5. a) Alter the cryptography file so that your alphabet minimally includes both capital and small English letters along with a blank space, comma, period, colon and the 10 digits 0 through 9.Ã‚Â  The number of symbols in your alphabet must be prime, so add dummy symbols such as small letters from the Greek alphabet to get to the smallest prime number of symbols in your alphabet.Ã‚Â  Our messages will be the symbols capital English letters, the 10 digits 0 through 9 and blank space, comma, period and colon where the dummy symbols of the small English letters only show up in the encrypted message.

Also, this alteration should include changing the block size to 7 for encryption by matrix multiplication by the 7 x 7 matrix E gotten randomly using rand for the Seed 48197.

b) Using the alterations of part a, encode and encrypt the message "REMEMBER THAT YOUR FINAL EXAM IS DUE ON TUESDAY APRIL 21ST AT 5:30 PM IN THIS ROOM. PLEASE BRING A HARD COPY OF IT TO CLASS. IF FOR SOME REASON YOU CAN NOT MAKE IT TO CLASS ON THIS DATE, THEN EMAIL IT TO ME WITH ALL OUTPUT REMOVED IF THE FILE IS TOO LARGE."

Check that this has worked by decoding and decrypting the resulting encoded and encrypted message from above.

Ã‚Â

 Subject Mathematics Due By (Pacific Time) 03/21/2015 08:00 pm
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