1. **State the problem:** We have a compound whose mass decreases over time according to the formula $$M(t) = 18e^{-kt}$$ where $k$ is a constant. 2. **Given:** After 25 minutes, the mass is 7.2 g. We need to show that $$18e^{-25k} = 7.2$$ and find $k$. 3. **Step (a)(i):** Substitute $t=25$ and $M(25)=7.2$ into the formula: $$18e^{-25k} = 7.2$$ 4. **Solve for $k$:** Divide both sides by 18: $$\frac{18e^{-25k}}{18} = \frac{7.2}{18}$$ $$\cancel{18}e^{-25k} = 0.4$$ 5. Simplify: $$e^{-25k} = 0.4$$ 6. Take the natural logarithm of both sides: $$\ln\left(e^{-25k}\right) = \ln(0.4)$$ $$-25k = \ln(0.4)$$ 7. Solve for $k$: $$k = -\frac{\ln(0.4)}{25}$$ 8. Calculate $k$ using a calculator: $$k = -\frac{-0.916290731874155}{25} = 0.0366516292749662$$ 9. Rounded to 5 decimal places: $$k = 0.03665$$ --- 10. **Step (a)(ii):** Find $t$ when $M(t) = 2.88$ g. 11. Substitute into the formula: $$18e^{-kt} = 2.88$$ 12. Divide both sides by 18: $$\frac{18e^{-kt}}{18} = \frac{2.88}{18}$$ $$\cancel{18}e^{-kt} = 0.16$$ 13. Simplify: $$e^{-kt} = 0.16$$ 14. Take natural logarithm: $$-kt = \ln(0.16)$$ 15. Solve for $t$: $$t = -\frac{\ln(0.16)}{k}$$ 16. Calculate $\ln(0.16)$: $$\ln(0.16) = -1.83258146374831$$ 17. Substitute $k=0.03665$: $$t = -\frac{-1.83258146374831}{0.03665} = 50.0$$ 18. Rounded to the nearest minute: $$t = 50$$ minutes **Final answers:** - $k = 0.03665$ - $t = 50$ minutes