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

Page 50 - Exercise 3B

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

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

Question 7

Part A

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