1. **Problem statement:** A spherical balloon has volume $V$ and radius $r$. Air is pumped in at rate $40\pi$ starting at $t=0$ with $r=0$, and air flows out at rate $0.8\pi r$. The balloon remains spherical. 2. **Volume formula:** The volume of a sphere is given by $$V = \frac{4}{3} \pi r^3$$ 3. **Relate rates:** Differentiate volume with respect to time $t$: $$\frac{dV}{dt} = 4 \pi r^2 \frac{dr}{dt}$$ 4. **Net volume rate:** Air pumped in minus air flowing out gives $$\frac{dV}{dt} = 40\pi - 0.8\pi r$$ 5. **Equate and solve for $\frac{dr}{dt}$:** $$4 \pi r^2 \frac{dr}{dt} = 40\pi - 0.8\pi r$$ Divide both sides by $4\pi r^2$: $$\frac{dr}{dt} = \frac{40\pi - 0.8\pi r}{4\pi r^2}$$ Cancel $\pi$: $$\frac{dr}{dt} = \frac{40 - 0.8 r}{4 r^2}$$ Divide numerator and denominator by 0.8: $$\frac{dr}{dt} = \frac{\frac{40}{0.8} - r}{\frac{4}{0.8} r^2} = \frac{50 - r}{5 r^2}$$ 6. **Final differential equation:** $$\boxed{\frac{dr}{dt} = \frac{50 - r}{5 r^2}}$$ This matches the required equation.