1. **State the problem:** We have a radioactive substance with an initial amount $y_0 = 35$ grams and a half-life of 230 years. We want to find the time $t$ when the amount $y$ decreases to 14 grams. 2. **Formula used:** The decay model is given by $$y = y_0 e^{-kt}$$ where $k$ is the decay constant. 3. **Find $k$ using the half-life:** The half-life $T_{1/2}$ satisfies $$\frac{y_0}{2} = y_0 e^{-k T_{1/2}}$$ Simplify: $$\frac{1}{2} = e^{-k \times 230}$$ Taking natural logarithm on both sides: $$\ln\left(\frac{1}{2}\right) = -230k$$ $$-\ln(2) = -230k$$ $$k = \frac{\ln(2)}{230}$$ Calculate $k$ rounded to 7 decimal places: $$k = \frac{0.6931472}{230} = 0.0030137$$ 4. **Find time $t$ when $y=14$ grams:** Substitute values into the decay formula: $$14 = 35 e^{-0.0030137 t}$$ Divide both sides by 35: $$\frac{14}{35} = e^{-0.0030137 t}$$ $$0.4 = e^{-0.0030137 t}$$ Take natural logarithm: $$\ln(0.4) = -0.0030137 t$$ $$-0.9162907 = -0.0030137 t$$ Solve for $t$: $$t = \frac{0.9162907}{0.0030137} = 304.07$$ 5. **Final answer:** It will take approximately **304.07 years** for the substance to decay from 35 grams to 14 grams.