1. **State the problem:** We want to find the year when women made up 30% of the labor force using the function $$f(x) = \frac{68.35}{1 + 0.036e^{-0.117x}}$$ where $x$ is the number of years since 1950. 2. **Set the function equal to 30:** $$30 = \frac{68.35}{1 + 0.036e^{-0.117x}}$$ 3. **Solve for the denominator:** $$1 + 0.036e^{-0.117x} = \frac{68.35}{30} = 2.2783$$ 4. **Isolate the exponential term:** $$0.036e^{-0.117x} = 2.2783 - 1 = 1.2783$$ 5. **Divide both sides by 0.036:** $$e^{-0.117x} = \frac{1.2783}{0.036} = 35.5083$$ 6. **Take the natural logarithm of both sides:** $$-0.117x = \ln(35.5083)$$ 7. **Calculate the logarithm:** $$\ln(35.5083) \approx 3.569$$ 8. **Solve for $x$:** $$x = \frac{-3.569}{0.117} = -30.5$$ 9. **Interpret $x$:** Negative $x$ means the year is before 1950. 10. **Calculate the year:** $$1950 + (-30.5) = 1919.5 \approx 1920$$ **Final answer:** Women made up 30% of the labor force around the year $\boxed{1920}$.