Question

the radius and circumference of several objects were measured. which best describes the strength of the correlation, and what is true about the causation between the variables? radius and circumference of objects it is a weak positive correlation, and it is not likely causal. it is a weak positive correlation, and it is likely causal. it is a strong positive correlation, and it is not likely causal. it is a strong positive correlation, and it is likely causal.

Explanation:

Step1: Recall the formula for circumference

The formula for the circumference \(C\) of a circle is \(C = 2\pi r\), where \(r\) is the radius.

Step2: Analyze the relationship

Since \(C = 2\pi r\), when \(r\) increases, \(C\) increases proportionally. The value of \(2\pi\approx6.28\).

• For \(r = 3\), \(C=2\pi\times3\approx 18.84\approx18.8\)
• For \(r = 4\), \(C = 2\pi\times4\approx25.12\approx25.1\)
• For \(r = 6\), \(C=2\pi\times6\approx37.68\approx37.7\)
• For \(r = 9\), \(C=2\pi\times9\approx56.52\approx56.5\)

The relationship is based on a mathematical formula, so it is a causal relationship. Also, since the formula shows a direct proportionality (strong linear - like relationship as \(C\) is a linear function of \(r\) in the form \(y = mx\) where \(y = C\), \(m = 2\pi\), \(x = r\)), the correlation is strong.

Answer:

It is a strong positive correlation, and it is likely causal.