# 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?

Question
Exponential growth and decay
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?

2021-02-09
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,
$$y(t)=a xx ekt$$ 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,
$$17.1=10.7*e^(k*4)$$
$$e^(4k)=log_e(17.1/10.7)=0.4688$$
k=0.1172
Hence, the equation of exponential growth rate is
$$y(t)10.7*e^(0.1172t)$$
Thus the annual growth rate is calculated at t=1 we get
$$y(1)=10.7*e^(0.1172*1)$$
$$y(1)=10.7*e^(0.1172)$$
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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