1. **State the problem:** We start with 350 grams of Turenchalkygen-62, which has a half-life of 170 days. We want to find how long it takes for the amount to decay to 90 grams. 2. **Formula used:** The decay of a radioactive substance is modeled by the formula: $$ A = A_0 \times \left(\frac{1}{2}\right)^{\frac{t}{T}} $$ where: - $A$ is the amount remaining after time $t$, - $A_0$ is the initial amount, - $T$ is the half-life, - $t$ is the time elapsed. 3. **Plug in known values:** $$ 90 = 350 \times \left(\frac{1}{2}\right)^{\frac{t}{170}} $$ 4. **Isolate the exponential term:** $$ \frac{90}{350} = \left(\frac{1}{2}\right)^{\frac{t}{170}} $$ 5. **Simplify the fraction:** $$ \frac{\cancel{90}}{\cancel{350}} = \frac{9}{35} $$ 6. **Take the natural logarithm of both sides:** $$ \ln\left(\frac{9}{35}\right) = \ln\left(\left(\frac{1}{2}\right)^{\frac{t}{170}}\right) $$ 7. **Use logarithm power rule:** $$ \ln\left(\frac{9}{35}\right) = \frac{t}{170} \times \ln\left(\frac{1}{2}\right) $$ 8. **Solve for $t$:** $$ t = 170 \times \frac{\ln\left(\frac{9}{35}\right)}{\ln\left(\frac{1}{2}\right)} $$ 9. **Calculate the logarithms:** $$ \ln\left(\frac{9}{35}\right) \approx \ln(0.2571) \approx -1.356 $$ $$ \ln\left(\frac{1}{2}\right) \approx -0.693 $$ 10. **Calculate $t$:** $$ t = 170 \times \frac{-1.356}{-0.693} = 170 \times 1.956 \approx 332.5 $$ **Final answer:** It takes approximately **332.5 days** for 350 grams of Turenchalkygen-62 to decay to 90 grams.