Khaleesi Herbert

2021-02-08

A "Student Drug Use and Health Survey" of Ontario high school students found that the percentage of students that reported serious psychosocial distress in the past month increased from 10.7% in 2013 to 17.1% in 2017.
Assuming a standard exponential growth trend, what is the annual growth rate in the percentage of students reporting serious psychosocial distress in the past month?

rogreenhoxa8

Let t=0 for 2013, t=1 for 2014, t=2 for 2015, t=3 for 2016, t=4 for 2017.
Assuming the exponential growth rate as,
$y\left(t\right)=axxekt$ where y is the % of students having serious psychosocial distress at time ta is value at starting t=0k is the rate of growtht is time
Therefore, at t=4, y=17.1
Thus,
$17.1=10.7\cdot {e}^{k\cdot 4}$
${e}^{4k}=lo{g}_{e}\left(\frac{17.1}{10.7}\right)=0.4688$
k=0.1172
Hence, the equation of exponential growth rate is
$y\left(t\right)10.7\ast {e}^{0.1172t}$
Thus the annual growth rate is calculated at t=1 we get
$y\left(1\right)=10.7\cdot {e}^{0.1172\ast 1}$
$y\left(1\right)=10.7\ast {e}^{0.1172}$
y(1)=12.03
Hence, the annual growth rate in the percentage of students reporting serious psychosocial distress in the past month is 12.03%.

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