5. Which measurement cannot be the circumference of a circle? Why? (
   

To determine which measurement cannot be the circumference of a circle, we need to consider the relationship between the circumference © and the radius ® of a circle. The formula for the circumference of a circle is:

where (pi) is approximately 3.14159.

Let’s examine the two given measurements:

For each measurement, we can rearrange the formula to solve for the radius:

Now, let’s calculate the radius for each circumference:

Both of these radii are positive real numbers, which means that both measurements could potentially be the circumference of a circle. However, the question asks which measurement “cannot” be the circumference of a circle. The key here is that the circumference of a circle must be a positive real number, as it represents the length of a curve.

Both and are positive real numbers, so neither of them inherently cannot be the circumference of a circle. However, if we consider the practicality of these measurements, we can infer that a circumference of is less likely to be a realistic measurement for a circle, as it would correspond to a very small circle with a radius of approximately .

Therefore, while neither measurement inherently “cannot” be the circumference of a circle, is less likely to be a realistic circumference due to the small size of the circle it would represent.