The Sixth Gate
After walking through the fifth gate, your computer informs you that more than half of the gates have been crossed. "Only 4 left", you conclude. "Computer, what is this gate's riddle?"

Your computer responds:

For the function, f(t)=1a(3t3−23t2+69)41
, find the value of a
that would make the slope of the tangent line −164
at t=2
.

a=

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Explain, in your own words and with your own work, how you arrived at this result. Be sure to explain using calculus concepts to best support the work of the game design team.