# Precalculus word problem on rocket height

I recently wrote a test in my precalculus class, and came across a problem which I thought I did correctly, but everyone else taking the course seems to have gotten a different solution. Because of the discrepancy, I was wondering if anyone would be willing to provide their own solution of the problem.

For what their worth, my thoughts on the problem: I thought that the rocket would be above $19.5$ metres after about $0.5$ seconds, and would never again fall below it, and I thought that part c) was a trick, since it never reaches a highest point, and never falls to the ground. As for the instantaneous rate of change, that I know is given by the derivative, which would be $H'(t) = 16t + 32$.)

And here is the problem:

A rocket fired straight up from the ground at a velocity of $32 \dfrac{m}{s}$, and the height above the ground is given by $H(t) = 8t^{2} + 32t$.

a) During what interval will the rocket be at least $19.5$ metres above the ground.

b) Calculate the average rate of change when the rocket reaches its highest point, to the time it falls to the ground.

c)Calculate the instantaneous rate of change at $t = 3$ $s$.

d)Explain why your answer to part b) and c) and positive and negative, respectively.

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Are you sure you didn't forget a minus sign in $H(t)$? $H(t) = -8t^2 + 32t$ looks like it would be more reasonable. – Scott Caldwell Dec 15 '12 at 20:00
I definitely didn't! That's why I'm so confused! I'm starting to suspect that the question was written incorrectly... – jack Dec 15 '12 at 20:16
What answer did the others taking the course arrive at? Could it be that they (expecting a downward-facing parabola) changed the problem to the one suggested by Scott? – Austin Mohr Dec 15 '12 at 20:26
Ahh, this is what I think probably happened: I missed the original test, so wrote it without the rest of my class. I think maybe someone corrected my teacher during the test, and they answered a revised question. she probably forgot to revise the question on my test. – jack Dec 15 '12 at 20:51
Not really relevant, but how are derivatives precalculus? – Henning Makholm Dec 15 '12 at 20:55