12: Section 4.2 Answers
- Page ID
- 103629
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)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