1. **State the problem:** We want to find the time $t$ when the amount of nitrogen pentoxide remaining, $N(t)$, is 2 grams. 2. **Recall the model:** The amount remaining at time $t$ is given by $$N(t) = 17 e^{-0.0005 t}$$ 3. **Set up the equation:** We want $N(t) = 2$, so $$2 = 17 e^{-0.0005 t}$$ 4. **Solve for $t$:** Divide both sides by 17: $$\frac{2}{17} = e^{-0.0005 t}$$ Take the natural logarithm of both sides: $$\ln\left(\frac{2}{17}\right) = -0.0005 t$$ Solve for $t$: $$t = -\frac{1}{0.0005} \ln\left(\frac{2}{17}\right)$$ 5. **Calculate the value:** Calculate the logarithm: $$\ln\left(\frac{2}{17}\right) = \ln(0.117647) \approx -2.1401$$ Then, $$t = -\frac{1}{0.0005} \times (-2.1401) = \frac{2.1401}{0.0005} = 4280.2$$ 6. **Interpretation:** It will take approximately 4280 seconds for the sample to decompose to 2 grams remaining. **Final answer:** $$t \approx 4280 \text{ seconds}$$