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

$ab^{3}=100\times 2^{3}=800{\mbox{ people}}$

Part B

$ab^{\frac {1}{2}}=100\times {\sqrt {2}}=141.41\dots \approx 141{\mbox{ people}}$

Part C

$ab^{\frac {7}{4}}=100\times {\sqrt[{4}]{2^{7}}}=336.35\dots \approx 336{\mbox{ people}}$

Question 3

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

Part A

$ab^{10}=35200\times 1.06^{10}=63037.8\dots \approx 63038$

Part B

$ab^{\frac {1}{2}}=35200\times {\sqrt {1.06}}=36240.61\approx 36241$

Part C

$ab^{-10}=35200\times {\frac {1}{1.06^{10}}}=19655.49\approx 19655$

Question 6

$a=440\,$

Part A

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

Part B

$ab^{-9}=440\times {\frac {1}{\sqrt[{12}]{2^{9}}}}=261.62\approx 262{\mbox{ Hz}}$

Part C

${\begin{matrix}ab^{x}=600\\\log a+x\log b=\log 600\\\log 440+x\log {\sqrt[{12}]{2}}=\log 600\\x\log {\sqrt[{12}]{2}}=\log 600-\log 440\\x={\frac {\log 600-\log 440}{\log {\sqrt[{12}]{2}}}}=5.36\dots \approx 5{\mbox{ whole semitones}}\end{matrix}}$

Question 7

Part A

${\begin{matrix}\log y=0.4+0.6x\\y=10^{0.4+0.6x}\\y=2.51\times 3.98^{x}\end{matrix}}$