1. **State the problem:** Solve for $t$ in the equation $5 = \frac{10.3}{1 + 12.8 e^{-0.036t}}$. 2. **Rewrite the equation:** Multiply both sides by the denominator to eliminate the fraction: $$5(1 + 12.8 e^{-0.036t}) = 10.3$$ 3. **Distribute 5:** $$5 + 64 e^{-0.036t} = 10.3$$ 4. **Isolate the exponential term:** $$64 e^{-0.036t} = 10.3 - 5$$ $$64 e^{-0.036t} = 5.3$$ 5. **Divide both sides by 64:** $$\cancel{64} e^{-0.036t} = \frac{5.3}{\cancel{64}}$$ $$e^{-0.036t} = \frac{5.3}{64}$$ 6. **Take the natural logarithm of both sides:** $$\ln\left(e^{-0.036t}\right) = \ln\left(\frac{5.3}{64}\right)$$ 7. **Simplify using the logarithm rule $\ln(e^x) = x$:** $$-0.036t = \ln\left(\frac{5.3}{64}\right)$$ 8. **Solve for $t$:** $$t = \frac{-\ln\left(\frac{5.3}{64}\right)}{0.036}$$ 9. **Calculate the numerical value:** $$\ln\left(\frac{5.3}{64}\right) = \ln(0.0828125) \approx -2.491$$ $$t = \frac{-(-2.491)}{0.036} = \frac{2.491}{0.036} \approx 69.2$$ **Final answer:** $$t \approx 69.2$$