2.

t(minutes) 0 2 5 7 11 12

r ' (t) (feet/min) 5.7 4.0 2.0 1.2 0.6 0.5 (Note: this is a table)

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 twice-differentiable function r of time t, where t is measured in minutes. For 0<t<12, the graph of r is concave down. The table above gives seleceted values of the rate of change, r ' (t), of the radius of the balloon over the time interval [0,12]. The radius of the balloon is 30 feet when t = 5. (Note: The volume of a sphere of radius r is given by V = 4/3(pi)(r^3).

a. Find the rate of change of the volume of the balloon with respect to time when t=5. Indicate units of measure.

b. Use a right reimann sum with the five subintervals indicated by the data in the table to approximate the integral(antidifferentiation) from 0 to 12 of r ' (t) dt.

c. Using the correct units, explain the meaning of your solution from part b in terms of the radius of the balloon.

Can't solve part a. It's asking for the rate of change of the "volume" with respect to time. Help!