User:Elboig

S______ L____

MATH 1314

Lab 3
Page 440 (117)

$$ \begin{align} D & = 5 e^{-0.4h} \\ (2) & = 5 e^{-0.4h} \\ 0.4 & = e^{-0.4h} \\ -0.4h & = ln0.4 \\ h & = \frac{ln0.4}{-0.4} \approx 2.29 \mbox{ hours.} \end{align} $$

Page 454 (11)

$$ \begin{align} ln x + ln(x + 2) & = 4 \\ ln((x + 2)(x)) & = 4 \\ ln(x^2 + 2 x) & = 4 \\ x^2 + 2 x & = e^4 \\ x^2 + 2 x - e^4 & = 0 \\ \end{align} $$

$$ \begin{align} x & = \frac{-b \pm \sqrt{b^2 - 4 ac}}{2a} \\ x & = \frac{-(2) \pm \sqrt{(2)^2 - 4 (1)(-e^4)}}{2(1)} \\ x & = \frac{-2 \pm \sqrt{4 + 4 e^4}}{2} \\ x & = \frac{-2 \pm \sqrt{4(1 + e^4)}}{2} \\ x & = \frac{-2 \pm 2\sqrt{1 + e^4}}{2} \\ x & = \frac{2(-1 \pm 1\sqrt{1 + e^4})}{2} \\ x & = -1 \pm \sqrt{1 + e^4} \\ \end{align} $$

$$ \begin{align} x & = -1 + \sqrt{1 + e^4} & x & = -1 - \sqrt{1 + e^4} \\ x & \approx 6.456 & x & \approx -8.456 \\ \end{align} $$

$$ \begin{align} \{ 6.456 \} \mbox { Since substituting -8.456 for x causes the natural logs to be undefined.} \\ \end{align} $$

Lab 6-7
Page 463 (31)

$$ \begin{align} A & = P (1+\frac{r}{n})^{nt} \\ 2P & = P (1+\frac{.08}{12})^{12 t}\\ 2 & = (1+\frac{.08}{12})^{12 t} \\ 2 & = (\frac{12.08}{12})^{12 t} \\ 12 t & = log_{\frac{12.08}{12}}2 \\ t & = \frac{log_{\frac{12.08}{12}}2}{12} \approx 8.69 \mbox{ years} \\ \end{align} $$

Page 463 (33)

(a)

$$ \begin{align} A & = P (1+\frac{r}{n})^{nt} \\ 150 & = 100(1+\frac{.08}{12})^{12 t} \\ 1.5 & = (\frac{12.08}{12})^{12 t} \\ log1.5 & = log(\frac{12.08}{12})^{12 t} \\ log1.5 & = 12 t \cdot log(\frac{12.08}{12}) \\ t & = \frac{log{1.5}}{12 log(\frac{12.08}{12})} & \approx 5.09 \mbox{ years} \\ \end{align} $$

(b)

$$ \begin{align} A & = Pe^{rt} \\ 150 & = 100 e^{.08 t} \\ 1.5 & = e^{.08 t} \\ .08 t & = ln{1.5} \\ t & = \frac{ln{1.5}}{.08} \approx 5.07 \mbox{ years} \\ \end{align} $$

Page 472(1)

(a)

$$ \begin{align} P(t) & = 500e^{0.02t} \\ P(0) & = 500e^{(0.02)(0)} = 500 \mbox { flies} \\ \end{align} $$

(b)

$$ \begin{align} & 0.02 \mbox{ or } 2\% \\ \end{align} $$

(c)

$$ \begin{align} P(10) & = 500 e^{(0.02)(10)} \approx 611 \mbox { flies} \\ \end{align} $$

(d)

$$ \begin{align} P(t) & = 500 e^{0.02 t} \\ 800 & = 500 e^{0.02 t} \\ 1.6 & = e^{0.02 t} \\ 0.02 t & = ln{1.6} \\ t & = \frac{ln{1.6}}{0.02} \approx 23.5 \mbox { days} \\ \end{align} $$

(e)

$$ \begin{align} P(t) & = 500 e^{0.02t} \\ 1000 & = 500 e^{0.02t} \\ 2 & = e^{0.02t} \\ 0.02 t & = ln{2} \\ t & = \frac{ln{2}}{0.02} \approx 34.7 \mbox{ days} \\ \end{align} $$

Page 472 (3)

(a)

$$ \begin{align} 0.0244 \mbox{ or } 2.44\% \\ \end{align} $$

(b)

$$ \begin{align} A(10) & = 500 e^{(-0.0244)(10)} \\ A(10) & = 500 e^{-0.244} \approx 391.744 \mbox{ grams}\\ \end{align} $$

(c)

$$ \begin{align} 400 & = 500 e^{-0.0244 t} \\ .8 & = e^{-0.0244 t} \\ -0.0244 t & = ln{.8} \\ t & = \frac{ln{.8}}{-0.0244} \approx 9.145 \mbox{ grams}\\

\end{align} $$

(d)

$$ \begin{align} 250 & = 500 e^{-0.0244 t} \\ .5 & = e^{.0.0244 t} \\ -0.0244 t & = ln.5 \\ t & = \frac{ln.5}{-0.0244} \approx 28.4 \mbox{ days}\\ \end{align} $$

Lab 8-9
Page 563 (77)

$$ \begin{align} 30 x + 15 y + 3 z & = 78 \\ 20 x + 2 y + 25 z & = 59 \\ 2 x + 20 y + 32 z & = 75 \\ \end{align} $$

$$ \begin{bmatrix} 30 & 15 & 3 & 78 \\   20 & 2 & 25 & 59 \\    2 & 20 & 32 & 75 \\  \end{bmatrix} $$

$$ \begin{bmatrix} 1 & 0 & 0 & 1.5 \\   0 & 1 & 0 & 2 \\    0 & 0 & 1 & 1 \\  \end{bmatrix} $$

$$ \begin{align} x & = 4000 \\ y & = 4000 \\ z & = 2000 \\ \end{align} $$

Page 563 (79)

Let $$x$$ = amount in T bills.

Let $$y$$ = amount in T bonds.

Let $$z$$ = amount in c bonds.

$$ \begin{align} x& + y& + &z && = 10000 \\ .06 x& + .07 y& + &.08 z && = 680 \\ x& & -& 2 z && = 0\\ \end{align} $$

$$ \begin{bmatrix} 1 & 1 & 1 & 10000 \\ .06 & .07 & .08 & 680 \\ 1 & 0 & -2 & 0 \\ \end{bmatrix} $$

$$ \begin{bmatrix} 1 & 0 & 0 & 4000 \\ 0 & 1 & 0 & 4000 \\ 0 & 0 & 1 & 2000 \\ \end{bmatrix} $$

$$ \begin{align} x & = 4000 \\ y & = 4000 \\ z & = 2000 \\ \end{align} $$

Page 590 (53)

$$ \begin{align} A & = \begin{bmatrix} 1 & -1 & 1 \\ 0 & -2 & 1 \\ -2 & -3 & 0 \\ \end{bmatrix} X & = \begin{bmatrix} x \\ y \\ z \\ \end{bmatrix} B & = \begin{bmatrix} 2 \\ 2 \\ .5 \\ \end{bmatrix} \end{align} $$

$$ \begin{align} A^{-1}B & = \begin{bmatrix} .5 \\ -.5 \\ 1 \end{bmatrix} \end{align} $$

$$ \begin{align} x & = 0.5 \\ y & = -0.5 \\ z & = 1 \\ \end{align} $$

Page 590 (57)

$$ A = \begin{bmatrix} 1 & 1 & 1 \\ 3 & 2 & -1 \\ 3 & 1 & 2 \\ \end{bmatrix} B = \begin{bmatrix} 2 \\ \frac{7}{3} \\ \frac{10}{3} \\ \end{bmatrix} $$

$$ a^{-1}B = \begin{bmatrix} \frac{1}{3} \\ 1 \\ \frac{2}{3} \\ \end{bmatrix} $$

$$ \begin{align} x & = \frac{1}{3} \\ y & = 1 \\ z & = \frac{2}{3} \\ \end{align} $$

Labs 4 & 10
Page 639 (13)

$$ \begin{align} a_n & = (-1)^{n + 1}n^2 \\ a_1 & = (-1)^{1 + 1}1^2 = 1 \\ a_2 & = ... = -4 \\ a_3 & = ... = 9 \\ a_4 & = ... = -16 \\ a_5 & = ... = 25 \\ \end{align} $$

Note: I made a simple program to help me on this one, adapting it for the next problem: Input x

((-1)^(x+1))x^2

Page 639 (15)

$$ \begin{align} a_n & = \frac{2^n}{3^n + 1} \\ a_1 & = \frac{2^1}{3^1 + 1} = \frac{2}{4} = \frac{1}{2} \\ a_2 & = ... = \frac{2}{5} \\ a_3 & = ... = \frac{2}{7} \\ a_4 & = ... = \frac{8}{41} \\ a_5 & = ... = \frac{8}{61} \\ \end{align} $$

Page 646 (39)

$$ \begin{align} a & = 2 \\ a_n & = 70 \\ d & = 2 \\ \end{align} $$

$$ \begin{align} n & = (\frac{a_n - a}{d}) + 1 \\ & = (\frac{70 - 2}{2}) + 1 \\ n & = 35 \\ \end{align} $$

$$ \begin{align} S_n & = \frac{n}{2}(a+a_n) \\ S_{35} & = \frac{35}{2}(2+70) \\ S_{35} & = 1260 \\

\end{align} $$

Page 646 (41)

$$ \begin{align} a & = 5 \\ a_n & = 49 \\ d & = 4 \\ \end{align} $$

$$ \begin{align} n & = (\frac{a_n - a}{d}) + 1 \\ n & = (\frac{49 - 5}{4}) + 1 \\ n & = 12 \\ \end{align} $$

$$ \begin{align} S_n & = \frac{n}{2}(a + a_n) \\ S_{12} & = \frac{12}{2}(5 + 49) \\ S_{12} & = 324 \\ \end{align} $$

Labs 11 & 12
Page 657 (59)

$$ \begin{align} a & = 1 \\ r & = \frac{1}{3} \\ \mbox { } \\ S & = \frac{a}{1-r} = \frac{1}{1-\frac{1}{3}} = \frac{1}{\frac{2}{3}} = 1 \cdot \frac{3}{2} = \frac{3}{2}\\ \end{align} $$

Page 657 (61)

$$ \begin{align} a & = 8 \\ r & = \frac{1}{2} \\ \mbox { } \\ S & = \frac{a}{1-r} = \frac{8}{1-\frac{1}{2}} = \frac{8}{\frac{1}{2}} = 8 \cdot \frac{2}{1} = 16 \\ \end{align} $$