Problem Four: 1998-1999

    An albatross is suddenly dropped into the sweltering atmosphere of Mercury. Due to the incredible heat and various gases of the planet, the bird quickly begins to burn to a crisp. If the position of the fire as it advances across the bird's body (oddly enough in a straight line from beak to butt) is determined by s(x) = (3x2 + 6x)4, find:

    (a)    The velocity and acceleration functions of the spreading flames.

    (b)    The time it takes for the bird to become completely charred if it is 16 feet long (that's a lotta bird!)

    (c)    The velocity and acceleration of the flames at that instant.

    (d)    How did an albatross get on Mercury anyhow?

Solution:
    (a)    v(x) = s'(x) = 4(3x2 + 6x)3*(6x + 6) = (24x + 24)(3x2 + 6x)3
        a(x) = v'(x) = (24x+24)(3)(3x2+6x)2(6x + 6) + (3x2 + 6x)3(24)
    Velocity is found using the Power Rule (and the Chain Rule subsequently).
    Acceleration is found using Product Rule (with a little Chain Rule sprinkled in).

    (b)    We want to know when the position of the flame reaches the end of the bird (or 16 feet), so we set the position equation equal to 16:

(3x2 + 6x)4=16
3x2 + 6x = 2
3x2 + 6x 2 = 0
Now, use the quadratic formula to get the solutions are x = .291 seconds and -2.291 sec. Clearly, the negative answer is ignored, so x = .291 sec.

    (c)    Remember, don't plug .291 into these equations!! Use the unrounded answer your calculator gave when you plug in (.2909944487), or your answer will likely be inaccurate.
    v(.291) = 247.871 ft/sec (don't forget units)
    a(.291) = 3072 ft/sec2

    (d)    There are two acceptable answers: either it was shot out of a cannon, or it's a super-evolved solid gold albatross.


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