

1. $$Q=20e^{-(t\ln 2)/3200}$$

2. $$\frac{2\ln 10}{\ln 2}\text{days}$$

3. $$\tau = 10\frac{\ln 2}{\ln 4/3}\text{minutes}$$

4. $$\tau\frac{\ln (p_{0}/p_{1})}{\ln 2}$$

5. $$\frac{t_{p}}{t_{q}}=\frac{\ln p}{\ln q}$$

6. $$k=\frac{1}{t_{2}-t_{1}}\ln \frac{Q_{1}}{Q_{2}}$$

7. $$20\text{ g}$$

8. $$\frac{50\ln 2}{3}\text{yrs}$$

9. $$\frac{25}{2}\ln 2%$$

10.

1. $$=20\ln 3\text{yr}$$
2. $$Q_{0}=100000e^{-5}$$

11.

1. $$Q(t)=5000-4750e^{-t/10}$$
2. $$5000\text{lbs}$$

12. $$\frac{1}{25}\text{ yrs}$$

13. $$V=V_{0}e^{t\ln 10/2} \: 4\text{ hours}$$

14. $$\frac{1500\ln \frac{4}{3}}{\ln 2}\text{yrs};2^{-4/3}Q_{0}$$

15. $$W(t)=20-19e^{-t/20};\lim_{t\to\infty }W(t)=20\text{ ounces}$$

16. $$S(t)=10(1+e^{-t/10};\lim_{t\to\infty}S(t)=10\text{ g}$$

17. $$10\text{ gallons}$$

18. $$V (t) = 15000 + 10000e ^{t/20}$$

19. $$W(t) = 4 × 10^{6} (t + 1)^{2}$$ dollars $$t$$ years from now

20. $$p=\frac{100}{25-24e^{-t/2}}$$

21.

1. $$P(t)=1000e^{.06t}+50\frac{e^{.06t}-1}{e^{.06/52}-1}$$
2. $$5.64\times 10^{-4}$$

22.

1. $$P-=rP-12M$$
2. $$P=\frac{12M}{r}(1-e^{rt})+P_{0}e^{rt}$$
3. $$M\approx\frac{rP_{0}}{12(1-e^{-rN})}$$
4. For (i) approximate $$M = 402.25$$, exact $$M = 402.80$$ for (ii) approximate $$M = 1206.05$$, exact $$M = 1206.93$$.

23.

1. $$T(\alpha )=-\frac{1}{r}\ln (1-(1-e^{-rN})/\alpha ))\text{ years}$$ $$S(\alpha )=\frac{P_{0}}{(1-e^{-rN})}[rN+\alpha\ln (1-(1-e^{-rN})/\alpha )]$$
2. $$T(1.05) = 13.69 \text{yrs}, S(1.05) = 3579.94 T(1.10) = 12.61 \text{yrs}, S(1.10) = 6476.63 T(1.15) = 11.70 \text{yrs}, S(1.15) = 8874.$$

24. $$P_{0}=\left\{\begin{array}{cc}{\frac{S_{0}(1-e^{(a-r)T})}{r-a}}&{\text{if }a\neq r}\\{S_{0}T}&{\text{if }a=r}\end{array} \right.$$