Question

15. In 2008, approximately 131 million people voted in the U.S. general election, compared to about 139 million people in 2016. The total population of the U.S. was 304 million in 2008 and 323 million in 2016. Assume that both values are growing exponentially. Find the growth rate r for both populations. State your answers as percentages, rounded to the nearest tenth of a percent.For the number of voters: r = %For the total population: r = %

Answer 1

`\begin{gathered} \text{Solve for the rate for number of voters:} \\ \text{At 2008,} \\ t=0 \\ y(0)=131,\text{ (in millions}) \\ \text{Solve for }a \\ y(t)=ae^(rt) \\ y(0)=ae^(r(0)) \\ 131=ae^0 \\ 131=a(1) \\ 131=a\rightarrow a=131 \\ \text{Substitute at year 2016, where }t=8\text{ (8 years has passed) and }y(8)=139\text{ (in millions)} \\ \text{Solve for }r \\ y(t)=ae^(rt) \\ y(8)=(131)e^(r(8)) \\ 139=131e^(8r) \\ (139)/(131)=(131e^(8r))/(131) \\ e^(8r)=(139)/(131),\text{ get the natural logarithm of both sides} \\ \ln e^(8r)=\ln ((139)/(131)) \\ 8r\ln e=\ln ((139)/(131)) \\ 8r=\ln ((139)/(131)) \\ r=(\ln((139)/(131)))/(8) \\ r=0.007409576241 \\ \text{Convert r into percentage and we have} \\ r=0.007409576241\cdot100\% \\ r=0.7409576241\% \\ r=0.7\%\text{ (rounded off to tenths)} \\ \text{Therefore, the rate for the number of voters is 0.7\%} \end{gathered}`
`\begin{gathered} \text{Solve for the rate for population} \\ \text{With 2008 as a starting point for }t=0,\text{ we know that }a=304\text{ (in millions)} \\ \text{We can now solve with }t=8\text{ (8 years has passed), }y(8)=323\text{ (in millions) for }r \\ y(t)=ae^(rt) \\ y(8)=(304)e^(r(8)) \\ 323=304e^(8r) \\ (323)/(304)=(304e^(8r))/(304) \\ e^(8r)=(323)/(304),\text{ get the natural logarithm of both sides} \\ \ln e^(8r)=\ln (323)/(304) \\ 8r\ln e^{}=\ln (323)/(304) \\ 8r=\ln (323)/(304) \\ r=(\ln (323)/(304))/(8) \\ r=0.007578077727 \\ \text{convert to percentage} \\ r=0.007578077727\cdot100\% \\ r=0.7578077727\% \\ r=0.8\%\text{ (rounded to tenth of a percent)} \\ \text{Therefore, the rate for the population is 0.8\%} \end{gathered}`