Calculus
posted by Craig .
t: 0 2 5 7 11 12
r'(t): 5.7 4.0 2.0 1.2 0.6 0.5
The volume of a spherical hot air balloon expands as the air inside the balloon is heated. The radius of the balloon in feet is modeled by a twicedifferentiable function of r of time t, where t is measured in minutes. For 0<t<12, the graph of r is concave down. The table above give selected values of the rate of change, r'(t), of the radius of the balloon over the time interval 0<t<12. The radius of the balloon is 30ft when t=5.
a)est. radius of balloon when t=5.4 using tan line approximated at t=5. Why?
b)Find rate of change of the volume of the balloon w/ respect to time when t=5. what is the units of measure?

sorry about that: when t=0, r't=5.7;
t=2, r't=4; t=5, r't=2; t=7, r't=1.2; t=11, r't=.6; t=12, r't=.5 
i really don't know how to start this problem

I have not answered your question since I am somewhat puzzled by the wording of your question.
by <<The radius of the balloon in feet is modeled by a twicedifferentiable function of r of time t>> do you mean that r is a quadratic function of t, a cubic or what?
As I see it, I could differentiate a function any number of times, just because I might reach a derivative of zero does not mean I couldn't do it once more.
e.g y = x^3 + ...
y' = 3x^2 ....
y'' = 6x ....
y''' = 6
y'''' = 0
y ''''' = 0 
i'm guessing that twice differentiable means y''

agree, but what level is the function?
e.g. If I knew that y = ... is quadratic, I could say y = at^2 + bt + c and go from there, but how do I know it's not a cubic? 
It does not matter I think.
We are not fitting the entire data set with a function but doing linear, then quadratic, interpolation.
For part a use radius at 5 and dr/dt at 5
r(5.4) = r(5) + .4 r'(5)
r(5.4) = 30 + .4 (2)
r(5.4) = 30 + .8
r(5.4) = 30.8 
V = Volume = (4/3) pi r^3 in ft^3
DV/dt = (4/3) pi (3) r^2 dr/dt in ft^3/sec
at t = 5
dV/dt = 4 pi r^2 (2)
= 8 pi (30)^2 = 240 pi ft^3/second
NOTE  the rate of change of volume is the surface area, 4 pi r^2 times the rate of change of radius  think about it.