Question

73
\the x - files: algebra adventures\ is a fictitious take on the classic show \the x - files\ that follows a team of investigators who use algebraic equations and algebraic unknowns (usually represented by x) to solve mysterious and quirky cases. the marketing team noticed minimal viewer growth over the first several days, but the trend eventually grew exponentially after word spread across the mathematics community. the total number of views, in thousands, was recorded for 12 days after the first episodes release. the exponential function v(t) was used to model the trend after t days.
the marketing team used the exponential function to conclude that the first episode will have approximately 19,832 views after 16 days. is this conclusion valid and why?
v(t)=0.39e^{0.23t} with r^{2}=0.97
yes, it is valid since r^{2} is close to 1, and t = 16 is a safe extrapolation.
yes, it is valid since the relationship follows an exponential trend.
no, this is not valid since v(16) is not 19,832.
no, this is not valid since the trend does not continue past t = 14.5.

Explanation:

Step1: Analyze the data - range

The data was recorded for 12 days. The graph shows the trend up to around \(t = 14.5\) days. Extrapolating to \(t = 16\) days is outside the observed - data range.

Step2: Consider the \(r^{2}\) value

The \(r^{2}=0.97\) indicates a strong correlation within the observed data range, but it doesn't guarantee the trend will continue beyond the observed range.

Answer:

No, this is not valid since the trend does not continue past \(t = 14.5\).