11.28: A.4.2- Section 4.2 Answers

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

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

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

4.

1. \(\approx 91.30^{\circ}\text{F}\)
2. \(8.99\) minutes after being placed out
3. never

5.

1. \(12:11:32\)
2. \(12:47:33\)

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.

1. \(Q'+\frac{2}{25}Q=6-2e^{-t/25}\)
2. \(Q=75-50e^{-t/25}-25e^{-2t/25}\)
3. \(75\)

16.

1. \(T=T_{m}+(T_{0}-T_{m})e^{-kt} +\frac{k(S_{0}-T_{m})}{(k-k_{m})}(e^{-kmt}-e^{-kt})\)
2. \(T=T_{m}+k(S_{0}-T_{m})te^{-kt}+(T_{0}-T_{m})e^{-kt}\)
3. \(\lim_{t\to\infty}T(t)=\lim_{t\to\infty}S(t)=T_{m}\)

17.

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

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. \(c_{1}=c(1-e^{-rt/W}),c_{2}=c(1-e^{-rt/W}-\frac{r}{W}te^{-rt/W})\)

20.

1. \(c_{n}=c\left(1-e^{-rt/W}\sum_{j=0}^{n-1}\frac{1}{j!}\left(\frac{rt}{W} \right)^{j} \right)\)
2. \(c\)
3. \(0\)

21. Let \(c_{\infty }=\frac{c_{1}W_{1}+c_{2}W_{2}}{W_{1}+W_{2}},\:\alpha =\frac{c_{2}W_{2}^{2}-c_{1}W_{1}^{2}}{W_{1}+W_{2}},\text{ and}\beta =\frac{W_{1}+W_{2}}{W_{1}W_{2}}.\) Then:

1. \(c_{1}(t)=c_{\infty }+\frac{\alpha }{W_{1}}e^{-r\beta t},c_{2}(t)=c_{\infty }-\frac{\alpha }{W_{2}}e^{-r\beta t}\)
2. \(\lim_{t\to\infty }c_{1}(t)=\lim_{t\to\infty }c_{2}(t)=c_{\infty }\)