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# A.4.2: Section 4.2 Answers

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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. $$\lim_{t\to\infty }c_{1}(t)=\lim_{t\to\infty }c_{2}(t)=c_{\infty }$$

This page titled A.4.2: Section 4.2 Answers is shared under a CC BY-NC-SA 3.0 license and was authored, remixed, and/or curated by William F. Trench.

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