User:Odd Bloke/C3/Chapter 3 - Exponential Growth and Decay

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Page 50 - Exercise 3B[edit]

Question 1[edit]

\begin{matrix}
100=ab^0 & \mbox{therefore} & a=100 \\
200=ab^1 & \mbox{therefore} & b=\frac{200}{100}=2
\end{matrix}

Part A[edit]


ab^3=100\times2^3=800\mbox{ people}

Part B[edit]


ab^{\frac{1}{2}}=100\times\sqrt{2}=141.41\dots\approx141\mbox{ people}

Part C[edit]


ab^{\frac{7}{4}}=100\times\sqrt[4]{2^7}=336.35\dots\approx336\mbox{ people}

Question 3[edit]


\begin{matrix}
ab^0=35200 & \mbox{therefore} & a=35200 \\
           &                  & b=1.06
\end{matrix}

Part A[edit]


ab^{10}=35200\times1.06^{10}=63037.8\dots\approx63038

Part B[edit]


ab^{\frac{1}{2}}=35200\times\sqrt{1.06}=36240.61\approx36241

Part C[edit]


ab^{-10}=35200\times\frac{1}{1.06^{10}}=19655.49\approx19655

Question 6[edit]


a=440\,

Part A[edit]


\begin{matrix}
ab^{12} = 2a           \\
b^{12}  = 2            \\
b       = \sqrt[12]{2} 
\end{matrix}

Part B[edit]


ab^{-9}=440\times\frac{1}{\sqrt[12]{2^9}}=261.62\approx262\mbox{ Hz}

Part C[edit]


\begin{matrix}
ab^x=600                          \\
\log a+x\log b=\log600            \\
\log440+x\log\sqrt[12]{2}=\log600 \\
x\log\sqrt[12]{2}=\log600-\log440 \\
x=\frac{\log600-\log440}{\log\sqrt[12]{2}}=5.36\dots\approx5\mbox{ whole semitones}
\end{matrix}

Question 7[edit]

Part A[edit]


\begin{matrix}
\log y=0.4+0.6x \\
y=10^{0.4+0.6x} \\
y=2.51\times3.98^x
\end{matrix}

Part B[edit]

Part C[edit]

Part D[edit]

Part E[edit]