9.3: Chapter 3 Answers

Section 3.1

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. a. \(=20\ln 3\text{yr}\)
b. \(Q_{0}=100000e^{-5}\)

11. a. \(Q(t)=5000-4750e^{-t/10}\)
b. \(5000\text{lbs}\)

12. \(\frac{1}{25}\text{ yrs}\)

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

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

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

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

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. \(p(t)={1000\over 1+999e^{-.9906t}}\) and \(p(6)=276\) people

22. \(5769\) fish

23. \(1400\) monkeys

24. \(1185\) otters

Section 3.2

1. \(\approx 15.15^{\circ}\text{F}\)

2. \(T=-10+110e^{-t\ln\frac{11}{9}}\)

3. \(\approx 24.33^{\circ}\text{F}\)

4. a. \(\approx 91.30^{\circ}\text{F}\)
b. \(8.99\) minutes after being placed outs
c. never

5. a. \(12:11:32\)
b. \(12:47:33\)

6. \((85/3)^{\circ}\text{C}\)

7. \(32^{\circ}\text{F}\)

8. \(Q(t)=40(1-e^{-3t/40})\)

9. \(Q(t)=30-20e^{-t/10}\)

10. \(K(t)=.3-.2e^{-t/20}\)

11. \(Q(50)=47.5\text{ (pounds)}\)

12. \(50\text{ gallons}\)

13. \(\text{min }q_{2}=q_{1}\sqrt{c}\)

14. \(Q=t+300-\frac{234\times 10^{5}}{(t+300)^{2}},\quad 0\leq t\leq 300\)

15. a. \(Q'+\frac{2}{25}Q=6-2e^{-t/25}\)
b. \(Q=75-50e^{-t/25}-25e^{-2t/25}\)
c. \(75\)

16. a. \(T=T_{m}+(T_{0}-T_{m})e^{-kt} +\frac{k(S_{0}-T_{m})}{(k-k_{m})}(e^{-kmt}-e^{-kt})\)
b. \(T=T_{m}+k(S_{0}-T_{m})te^{-kt}+(T_{0}-T_{m})e^{-kt}\)
c. \(\lim_{t\to\infty}T(t)=\lim_{t\to\infty}S(t)=T_{m}\)

17. a. \(T'=-k(1+\frac{a}{a_{m}})T+k(T_{m0}+\frac{a}{a_{m}}T_{0})\)
b. \(T=\frac{aT_{0}+a_{m}T_{m0}}{a+a_{m}}+\frac{a_{m}(T_{0}-T_{m0})}{a+a_{m}}e^{-k(1+a/a_{m})t},\quad T_{m}=\frac{aT_{0}+a_{m}T_{m0}}{a+a_{m}}+\frac{a(T_{m0}-T_{0})}{a+a_{m}}e^{-k(a+a/a_{m})t}\)
c. \(\lim_{t\to\infty }T(t)=\lim_{t\to\infty}T_{m}(t)=\frac{aT_{0}+a_{m}T_{m0}}{a+a_{m}}\)

18. \(V=\frac{a}{b}\frac{V_{0}}{V_{0}-(V_{0}-a/b)e^{-at}};\quad\lim_{t\to\infty }V(t)=a/b\)

19. \(P(t)={10,000,000\over (1000+t)^2};\quad\) \(t=2162\) days, or almost \(6\) years

Section 3.3

1. \(v=-\frac{384}{5}(1-e^{-5t/12});\quad -\frac{384}{5}\text{ft/s}\)

2. \(k=12;\quad v=-16(1-e^{-2t})\)

3. \(v=25(1-e^{-t});\quad 25\text{ft/s}\)

4. \(v=20-27e^{-t/40}\)

5. \(\approx 17.10\text{ ft}\)

6. \(v=-\frac{40(13+3e^{-4t/5})}{13-3e^{-4t/5}};\quad -40\text{ft/s}\)

7. \(v=-128(1-e^{-t/4})\)

8. \(v=-\frac{64(1-e^{-t})}{1+e^{-t}};\quad -64\text{ft/s}\)

9. \(v=\alpha\frac{v_{0}(1+e^{-\beta t})-\alpha (1-e^{-\beta t})}{\alpha (1+e^{-\beta t})-v_{0}(1-e^{-\beta t})};\quad -\alpha\) where \(\alpha =\sqrt{\frac{mg}{k}}\) and \(\beta =2\sqrt{\frac{kg}{m}}\)

10. \(\approx 6.93\text{miles/s}\)

11. \(\approx 1.47\text{miles/s}\)