Related Rates Under a Constraint: The Expanding Balloon
A spherical balloon is being inflated so that its volume increases at a constant 100 cm^3/s. A technician measures the radius when it reaches 5 cm and wants to know how fast the radius is growing at that instant. A second technician insists the radius grows at a constant rate too, since the volume rate is constant. A third points out that dV/dt is constant but dr/dt depends on r, so the radius growth slows as the balloon gets bigger. The team must resolve the disagreement, and the lead asks them to also compute the rate of change of the surface area at the same instant, and to say whether the surface-area rate is constant.
1. Find dr/dt when r=5 cm, given dV/dt=100 cm^3/s and V=(4/3)pi r^3.
Read the worked answer
Differentiate V=(4/3)pi r^3 with respect to t: dV/dt = 4 pi r^2 (dr/dt).
Solve: dr/dt = (dV/dt)/(4 pi r^2) = 100/(4 pi * 25) = 100/(100 pi) = 1/pi cm/s.
So dr/dt = 1/pi ≈ 0.318 cm/s at r=5 cm. This is a snapshot value; because the denominator contains r^2, dr/dt decreases as r grows even though dV/dt is constant. That resolves the dispute: the volume rate is constant, the radius rate is not.
2. Compute dA/dt when r=5 cm, where A=4 pi r^2, and determine whether dA/dt is constant as the balloon inflates.
Read the worked answer
A=4 pi r^2, so dA/dt = 8 pi r (dr/dt).
At r=5 with dr/dt=1/pi: dA/dt = 8 pi * 5 * (1/pi) = 40 cm^2/s.
To check constancy, substitute dr/dt = (dV/dt)/(4 pi r^2): dA/dt = 8 pi r * (dV/dt)/(4 pi r^2) = 2(dV/dt)/r = 200/r. Since r is increasing, dA/dt = 200/r is decreasing, not constant. At r=5 it is 40 cm^2/s; at larger r it is smaller. So the surface-area rate is not constant.
3. A teammate claims that because dV/dt is constant, d^2V/dt^2 = 0 and therefore d^2r/dt^2 = 0. Identify the error.
Read the worked answer
The error is assuming second derivatives pass through the chain rule unchanged. dV/dt constant means d^2V/dt^2 = 0, true. But dr/dt = (dV/dt)/(4 pi r^2) is a function of r(t), so differentiating again: d^2r/dt^2 = (dV/dt) * d/dt[(4 pi r^2)^{-1}] = (dV/dt)(-2)(4 pi r^2)^{-2}(8 pi r)(dr/dt), which is negative for r>0 and dr/dt>0. Concretely d^2r/dt^2 = -(dV/dt)^2/(8 pi^2 r^5), nonzero. The relation between V and r is nonlinear, so a constant first derivative of V does not imply a constant first derivative of r. The teammate conflated 'dV/dt constant' with 'all rates constant.'