Question

The rate of change of the altitude of a hot air balloon is given by:
r(t)= t^3 - 4t^2 + 6
for 0 is less than or equal to t and less than or equal to 8.

its asking which of the following expressions gives the change in altitude during the time the altitude is decreasing.
I'm stuck between the answer -
8
(integral sign) r(t) dt.
0
I need help with the steps. I kind of have an idea of how to set it up but I'm not sure.

Answer 1

They want to know where is the function negative

Answer 2

find criticle point by setting it equal to zero

Answer 3

I don't think you need integral at all

Answer 4

Hmmm thank you :) still trying to work it out. not sure if I have quite gotten it yet.

Answer 5

Still didn't get the answer "[ I think I did it backwards because i got a positive answer lol

Answer 6

you answer has to be between 0 and 8

Answer 7

i got \[\int\limits_{0}^{2.667?} r(t) dt\] it is one of my choices but I don't think I'm right.

Answer 8

t^3 - 4t^2 + 6=0

solve to t, I will let wolfram do it

1.57 to 3.51 is out goal(negative value)

\[\int_{1.57}^{3.51} r(t)dt\]

Answer 9

that's one of the choices....To check I integrated all the choices to see which would come out negative, 3 of them did. one being your choice, and the integral of r'(t) with the same interval. I got kind of stuck, and i was trying to find a different way to do it, and I've gotten different answers :/

Answer 10

The interval I gave is entire time the balloon is in descent.(negative change all the whole time). The other two which integrated to be negative are only be partially in descent.(it turned out to be negative because balloon descent more than ascent in those inteval)

Answer 11

ahhh oh ok. I see what you are saying.