Question

scatterplot a
volume over time
scatterplot b
volume over time
scatter
volume ov
which model best represents the relationship between time and volume?
a linear model would best represent the relationship because scatterplot a is fairly linear.
an exponential model would best represent the relationship because scatterplot b is fairly linear.
a power model would best represent the relationship because scatterplot c is fairly linear.
a linear model would best represent the relationship because scatterplot c is fairly linear.

Explanation:

Step1: Recall model - transformation relationships

For an exponential model \(y = ab^{x}\), taking the logarithm of both sides gives \(\log(y)=\log(a)+x\log(b)\). If we let \(Y = \log(y)\), then \(Y\) is a linear function of \(x\) (\(Y = c + mx\), where \(c=\log(a)\) and \(m = \log(b)\)).

Step2: Analyze scatter - plots

Scatterplot A (with \(y\) - axis as volume and \(x\) - axis as time) is not linear. Scatterplot B has \(Y=\log(\text{volume})\) on the \(y\) - axis and \(x\) (time) on the \(x\) - axis. Since scatterplot B is fairly linear, and we know that for an exponential model \(y = ab^{x}\), \(\log(y)\) is a linear function of \(x\).

Answer:

An exponential model would best represent the relationship because scatterplot B is fairly linear.