Final answer:
To find the rate at which the radius of the spherical balloon is increasing, we use the formula V = (4/3)πr³ and differentiate it with respect to time, given a volume increase of 30 cm³/min and a volume of 80π cm³. Solving for the radius first, we then calculate the derivative to find the rate of radius increase.
Step-by-step explanation:
The student is asking about the rate of change of the radius of a spherical balloon as it increases in volume. To solve this, we use the formula for the volume of a sphere V = (4/3)πr³. Given the volume of the balloon is 80π cm³, we can solve for the radius r first, which will then allow us to use related rates to find how quickly the radius is changing when the volume is increasing at 30 cm³/min.
First, equate the given volume to the volume formula and solve for r:
- 80π = (4/3)πr³
- 60 = r³
- r = ∛60 ≈ 3.9149 cm
To find the rate of change of the radius, dr/dt, we differentiate both sides of the volume equation with respect to time t:
- dV/dt = (4/3)π × 3r² × dr/dt
- 30 = 4πr² × dr/dt
- dr/dt = 30 / (4πr²)
Plugging the value of r from above into this equation gives us the rate of change of the radius at that instant.