1. **State the problem:** Given the rate of growth of Rapunzel's hair as $$\frac{dL}{dt} = \frac{1}{5t}$$, find the length of hair grown between the 100th day and the 200th day. 2. **Understand the problem:** The function $$\frac{dL}{dt}$$ represents the rate of change of hair length with respect to time. To find the length grown in the interval from $$t=100$$ to $$t=200$$, we need to integrate the rate function over this interval. 3. **Set up the integral:** $$L = \int_{100}^{200} \frac{1}{5t} \, dt$$ 4. **Calculate the integral:** Rewrite the integral: $$L = \frac{1}{5} \int_{100}^{200} \frac{1}{t} \, dt$$ The integral of $$\frac{1}{t}$$ is $$\ln|t|$$, so: $$L = \frac{1}{5} [\ln|t|]_{100}^{200} = \frac{1}{5} (\ln 200 - \ln 100)$$ 5. **Simplify the logarithmic expression:** $$\ln 200 - \ln 100 = \ln \left( \frac{200}{100} \right) = \ln 2$$ So, $$L = \frac{1}{5} \ln 2$$ 6. **Evaluate the numerical value:** Using $$\ln 2 \approx 0.693$$, $$L = \frac{1}{5} \times 0.693 = 0.1386$$ meters. 7. **Interpret the result:** The hair length grown between the 100th and 200th days is approximately 0.139 meters, which corresponds to option B. **Final answer:** $$\boxed{0.139 \text{ m}}$$