Air is pumped into a spherical balloon so that its volume increases at a constant rate of 100 cm^3/s. At what rate is the radius of the balloon increasing when the radius is 5.0 cm?
Answer and explanation
Correct answer: D. 1 / (pi) cm/s (approximately 0.318 cm/s)
The volume of a sphere is V = (4/3) pi r^3. Differentiating with respect to time: dV/dt = 4 pi r^2 (dr/dt). Given dV/dt = 100 cm^3/s and r = 5.0 cm: 100 = 4 pi (5.0)^2 (dr/dt) => 100 = 4 pi 25 (dr/dt) => 100 = 100 pi * (dr/dt). Solving yields dr/dt = 1 / pi cm/s ≈ 0.318 cm/s.
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