The Enigma of the Cosmic Journey

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A space probe travels along a path described by the function

�(�)=��3+��2+��+�, where �, �, �, and � are constants. The probe starts its journey at �=0 and reaches its furthest point at �=4 hours, where it then changes its course. Given that the probe's path reaches a maximum height at �=2 hours and the heights at �=0, �=2, and �=4 are 0, 64, and 0 meters respectively, determine the values of �, �, �, and �.

Hint: Use the conditions given to set up equations and solve for the unknowns.

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