1. Part (a)
This estimate of 30.8’ is HIGHER than the actual value because r(t) is CONCAVE DOWN.
(r - ) = (t - ) 30 5 2
r = 2t + 20
r  2(5.4) + 20
r  30.8 feet

2. Part (b)
Find dV/dt when t=5:
Volume = 4/3 p r3
dV/dt = 4 p r2 (dr/dt)
dV/dt = 4 p (30)2 (2)
dV/dt = 7200 p ft3/min

3. (2)(4) + (3)(2) + (2)(1.2) + (4)(.6) + (1)(.5) 8 + 6 + 2.4 + 2.4 + .5
Part (c) 2 5 7 11 12
(0,5.7)
(2,4)
(5,2)
(7,1.2)
(11,0.6)
(12,0.5)
Right Riemann sum:
r’(t) dt  12
(2)(4) + (3)(2) + (2)(1.2) + (4)(.6) + (1)(.5)
This represents the total change in the balloon’s radius during the 12 minute interval.
 19.3

4. (IGNORED) Part (d) r’(t) dt.2 5 7 11 12
(0,5.7)
(2,4)
(5,2)
(7,1.2)
(11,0.6)
(12,0.5)
(IGNORED)
r’(t) dt. 12
Our approximation of 19.3 feet was less than
Since the curve was decreasing, we purposely ignored these white areas that should have been considered.