Delicious The following table shows the popularity of...

Delicious The following table shows the popularity of Delicious among social media sites as rated by StatCounter.com ( $t$ is the number of years since the start of 2008 ): $$\begin{array}{|r|c|c|c|c|c|}\hline \text { Year } t & 1 & 2 & 3 & 4 & 5 \\\hline \text { year since start of } 2008) & & & & & \\\hline \begin{array}{r}\text { Delicious Popularity } p(t) \\(\%)\end{array} & 0.4 & 0.2 & 0.1 & 0.05 & 0.02 \\\hline\end{array}$$ a. Represent $p$ graphically, and then use your graph to estimate $p(3.5)$. Interpret your answer. b. One of the following models fits the data exactly. Which model is it? (A) $p(t)=0.8\left(2^{-t}\right)$ (B) $p(t)=0.8\left(2^{\prime}\right)$ (C) $p(t)=0.02 t^{2}-0.2 t+0.6$ (D) $p(t)=-0.02 t^{2}+0.2 t-0.6$

Delicious The following table shows the popularity of Delicious among social media sites as rated by StatCounter.com ( $t$ is the number of years since the start of 2008 ):
$$\begin{array}{|r|c|c|c|c|c|}\hline \text { Year } t & 1 & 2 & 3 & 4 & 5 \\\hline \text { year since start of } 2008) & & & & & \\\hline \begin{array}{r}\text { Delicious Popularity } p(t) \\(\%)\end{array} & 0.4 & 0.2 & 0.1 & 0.05 & 0.02 \\\hline\end{array}$$
a. Represent $p$ graphically, and then use your graph to estimate $p(3.5)$. Interpret your answer.
b. One of the following models fits the data exactly. Which model is it?
(A) $p(t)=0.8\left(2^{-t}\right)$
(B) $p(t)=0.8\left(2^{\prime}\right)$
(C) $p(t)=0.02 t^{2}-0.2 t+0.6$
(D) $p(t)=-0.02 t^{2}+0.2 t-0.6$

Key Concepts

Exponential Decay

Exponential decay describes a process where a quantity decreases at a rate proportional to its current value. This type of behavior is common in many natural and social phenomena, from radioactive decay to declines in popularity or market share over time. Recognizing an exponential decay pattern is key in selecting the appropriate mathematical model and understanding the underlying dynamics of the data.

Graphical Representation

Graphical representation involves plotting the given data points on a coordinate plane to visually identify trends or patterns in the data. In a broader context, this method is essential for understanding the behavior of functions and their relationships without relying solely on numerical data. It facilitates qualitative analysis and helps in making conjectures about the underlying mathematical relationships.

Interpolation

Interpolation is the process of estimating unknown values that fall between known data points on a graph. This concept is crucial in numerical analysis and data modeling, as it allows one to predict values within the range of the data given, even if the exact function is not immediately apparent. It is especially useful when evaluating functions at points where measurements or evaluations have not been explicitly provided.

Model Selection

Model selection is the process of choosing a mathematical model that best describes the behavior of a data set. It is a critical step in mathematical modeling and data analysis because the correct model enables accurate predictions and deeper insights into the phenomenon being studied. This often involves comparing different candidate models to see which one fits the data exactly or most closely, using both graphical and algebraic reasoning.

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