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

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

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

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

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

Question 3
$$ \begin{matrix} ab^0=35200 & \mbox{therefore} & a=35200 \\ &                 & b=1.06 \end{matrix} $$

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

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

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

Question 6
$$ a=440\, $$

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

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

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

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