Final answer:
The diameter of the balloon is increasing at a rate of approximately 0.435 feet per minute.
Step-by-step explanation:
To find how rapidly the diameter of the balloon is increasing, we need to determine the rate at which the radius is changing. We can use the formula for the volume of a sphere to relate the rate of change of volume with the rate of change of radius. The volume of a sphere is given by V = (4/3)πr³, where V is the volume and r is the radius. We can differentiate both sides of the equation with respect to time to find the relationship between the rates of change:
dV/dt = 4πr²(dr/dt)
Given that dV/dt = 3.1 ft³/min and the diameter is 1.5 feet, we can substitute these values into the equation and solve for dr/dt:
3.1 = 4π(0.75)²(dr/dt)
Simplifying,
3.1 = 2.25π(dr/dt)
dr/dt = 3.1/(2.25π)
dr/dt ≈ 0.435 ft/min
Therefore, the diameter of the balloon is increasing at a rate of approximately 0.435 feet per minute when the diameter is 1.5 feet.