T12.2 Students in a statistics class drew circles of varying diameters and counted how many Cheerios could be placed in the circle. The scatterplot shows the results. The students want to determine an appropriate equation for the relationship between diameter and the number of Cheerios. The students decide to transform the data to make it appear more linear before computing a least-squares regression line. Which of the following transformations would be reasonable for them to try?

I. Plot the square root of the number of Cheerios against diameter.
II. Plot the cube of the number of Cheerios against diameter.
III. Plot the log of the number of Cheerios against the log of the diameter.
IV. Plot the number of Cheerios against the log of the diameter.

a. I and II
b. I and III
c. II and III
d. II and IV
e. I and IV

To determine that which of the transformations would be reasonable.

Since the students decided to draw a circle and see how many cheerios they could fit inside it. They alter the data to make it appear more linear before computing a linear regression line to observe the better outcome.
As a result, we must determine which of the following transformations they should attempt.
As a result, we can see that the pattern in the scatterplot looks like a quadratic function, implying that a graph of the type $y=a{x}^2+b$could be a good model. However, this indicates that the number of cheerios is equal to the square of the diameter up to a certain number of constants, and thus the proper transformation is to take the square root of the number of cheerios.

As a result, the transformation I is appropriate.
Similarly, the pattern of the given scatterplot resembles that of an exponential function, suggesting that it could be a model.
However, this indicates that the number of cheerios is the exponential of the diameter up to a constant, and that the logarithm of the number of cheerios is the appropriate transformation.
As a result, III is a suitable transition.
As a result, option (b) I and III is the correct answer.