Question: Use linear approximation to estimate the amount of paint in cubic centimeters needed to apply a coat of paint $0.010000$ cm thick to a hemispherical dome with a diameter of $45.000$ meters.
1. **State the problem:** We want to estimate the volume of paint needed to coat a hemispherical dome of diameter $45.000$ meters with a paint thickness of $0.010000$ cm.
2. **Convert units:** Diameter is $45.000$ meters, so radius $r = \frac{45.000}{2} = 22.5$ meters.
Since paint thickness is in centimeters, convert radius to centimeters:
$$22.5 \text{ meters} = 22.5 \times 100 = 2250 \text{ cm}$$
3. **Volume of a sphere:**
$$V = \frac{4}{3} \pi r^3$$
Volume of hemisphere:
$$V_{hemi} = \frac{1}{2} V = \frac{2}{3} \pi r^3$$
4. **Linear approximation:**
We approximate the change in volume $\Delta V$ when radius increases by paint thickness $\Delta r = 0.010000$ cm.
The derivative of volume with respect to radius is:
$$\frac{dV}{dr} = \frac{d}{dr} \left( \frac{2}{3} \pi r^3 \right) = 2 \pi r^2$$
5. **Calculate $\Delta V$:**
$$\Delta V \approx \frac{dV}{dr} \times \Delta r = 2 \pi r^2 \times 0.010000$$
Substitute $r = 2250$ cm:
$$\Delta V = 2 \pi (2250)^2 \times 0.010000$$
6. **Calculate intermediate values:**
$$2250^2 = 5062500$$
So,
$$\Delta V = 2 \pi \times 5062500 \times 0.010000 = 2 \pi \times 50625$$
7. **Final calculation:**
$$\Delta V = 2 \times 3.141592653589793 \times 50625 \approx 318309.8862 \text{ cm}^3$$
8. **Interpretation:** The volume of paint needed is approximately $318309.8862$ cubic centimeters.
**Note:** The number $21205938.9077$ given in the prompt does not match this calculation and might be a different value or unrelated.
**Final answer:**
$$\boxed{318309.8862 \text{ cm}^3}$$