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The radius and circumference of several objects were measured.



\begin{table}

\captionsetup{labelformat=empty}

\caption{Radius and Circumference of Objects}


Radius (in.)		Circumference (in.)

3		18.8

4		25.1

6		37.7

9		56.5




\end{table}



Which best describes the strength of the correlation, and what is true about the causation between the variables?


AIt is a weak positive correlation, and it is not likely causal.
BIt is a weak positive correlation, and it is likely causal.
CIt is a strong positive correlation, and it is not likely causal.
DIt is a strong positive correlation, and it is likely causal.

Verified Answer :

  • The data points follow a consistent linear pattern where increasing the radius results in a proportional increase in circumference.
  • The ratio of \(\frac{C}{r}\) is consistently close to \(2\pi \approx 6.28\), indicating a strong positive correlation.
  • The geometric formula \(C = 2\pi r\) proves that the circumference is a direct consequence of the radius, making the relationship causal.
  • The correct description is: \(\boxed{\text{It is a strong positive correlation, and it is likely causal.}}\)

Explanation

  1. Analyze the data trend

To evaluate the relationship between the radius (\(r\)) and the circumference (\(C\)), we first analyze the given data points. As the radius increases from \(3\) to \(9\) inches, the circumference increases from \(18.8\) to \(56.5\) inches. This indicates a strong, consistent, and positive trend.

  1. Calculate the strength of the correlation

The mathematical relationship between the radius and circumference of a circle is defined by the formula \(C = 2\pi r\). This implies that the ratio \(\frac{C}{r}\) should be equal to \(2\pi\), which is approximately \(6.28\). Dividing the given circumferences by their corresponding radii, we find:

  • For \(r = 3\): \(\frac{18.8}{3} \approx 6.27\)
  • For \(r = 4\): \(\frac{25.1}{4} \approx 6.28\)
  • For \(r = 6\): \(\frac{37.7}{6} \approx 6.28\)
  • For \(r = 9\): \(\frac{56.5}{9} \approx 6.28\)

The ratios are extremely close to \(2\pi\), confirming a very strong positive linear correlation.

  1. Determine the causality of the relationship

A relationship is considered causal if one variable directly determines or influences the other. Geometry dictates that the radius of a circle mathematically determines its circumference through the formula \(C = 2\pi r\). Since the circumference is a direct function of the radius, the relationship is inherently causal. Therefore, we can conclude that there is a strong positive correlation and the relationship is likely causal.

Examples

Understanding the relationship between radius and circumference is fundamental in engineering and manufacturing. For example, when designing bicycle wheels or gear systems, knowing that the circumference (which determines distance traveled per rotation) is directly caused by the chosen radius allows engineers to design precise components for specific performance needs, such as gear ratios or speed optimization. Calculations involving \(C = 2\pi r\) ensure that components fit and function correctly within a larger assembly.

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