1. **State the problem:** We want to find the differential formulas that estimate the change in volume $dV$ of a sphere when its radius changes by a small amount $dr$ from an initial radius $r_0$. 2. **Recall the volume formula:** The volume of a sphere is given by $$V = \frac{4}{3}\pi r^3$$ 3. **Find the differential formula:** To estimate the change in volume $dV$, we differentiate $V$ with respect to $r$: $$\frac{dV}{dr} = 4\pi r^2$$ 4. **Write the differential:** The differential $dV$ is $$dV = \frac{dV}{dr} dr = 4\pi r^2 dr$$ 5. **Answer part (A):** Using $r_0$ as the initial radius, $$dV = 4\pi r_0^2 dr$$ 6. **Answer part (B):** When the radius changes from 8 to $8 + dr$, substitute $r_0 = 8$: $$dV = 4\pi (8)^2 dr = 256\pi dr$$ 7. **Answer part (C):** To estimate the change in volume when the radius changes from 8 cm to 7.8 cm, calculate $dr = 7.8 - 8 = -0.2$ cm. Substitute into the differential formula: $$dV = 256\pi (-0.2) = -51.2\pi$$ Approximate numerically: $$dV \approx -51.2 \times 3.1416 = -160.85 \text{ cm}^3$$ This means the volume decreases by approximately 160.85 cubic centimeters.