To solve this problem, we can use the formula for the volume of a sphere:
V = (4/3)πr³
where V is the volume and r is the radius of the balloon.
Given that the volume is increasing at a rate of 2.1 ft/min, we can express this as:
dV/dt = 2.1 ft/min
We need to find how rapidly the radius (dr/dt) is changing when the radius is 0.65 feet.
To relate the rate of change of volume to the rate of change of radius, we can differentiate the volume formula with respect to time (t):
dV/dt = (dV/dr) * (dr/dt)
We know dV/dt = 2.1 ft/min. Now, let's find (dV/dr):
V = (4/3)πr³
Differentiating both sides with respect to r:
dV/dr = 4πr²
Now, we can substitute the values into the formula:
2.1 ft/min = (4π(0.65)²) * (dr/dt)
Simplifying:
2.1 ft/min = 4π(0.4225) * (dr/dt)
Now, solve for (dr/dt):
(dr/dt) = (2.1 ft/min) / (4π(0.4225))
(dr/dt) ≈ 0.396 ft/min
Therefore, when the radius is 0.65 feet, the radius of the balloon is increasing at a rate of approximately 0.396 ft/min.