Final answer:
To find the rate at which the volume is increasing, we use the derivative of the volume formula with respect to time. Plugging in the rates of change for radius and height and the given dimensions into the derived formula, the volume is increasing at approximately 87.96452 cm³/s when the height is 9 cm and the radius is 4 cm.
Step-by-step explanation:
The question involves calculating the rate of change of volume for a cylinder, which is a problem in related rates, a topic in calculus. The volume of a cylinder is given by V = πr²h. When the radius r is changing at a rate of dr/dt = 1/3 cm/s and the height h is changing at a rate of dh/dt = 1/2 cm/s, we can use the chain rule to find the rate of change of volume, dV/dt.
Applying the derivative to both sides of the volume formula concerning time gives us:
dV/dt = π (2rh dr/dt + r² dh/dt)
Plugging in the given values, we get:
dV/dt = π (2 × 4 cm × 9 cm × (1/3) cm/s + (4 cm)² × (1/2) cm/s)
dV/dt = π (24 cm²/s + 8 cm² × (1/2) cm/s)
dV/dt = π (24 cm²/s + 4 cm²/s)
dV/dt = π × 28 cm²/s
dV/dt = 28π cm³/s
Substituting the value π ≈ 3.14159, we find:
dV/dt ≈ 28 × 3.14159 cm³/s
dV/dt ≈ 87.96452 cm³/s, which does not match any of the options provided in the question.