I don't think we need a comma before since. When since is used as a conjunction to mean "because" at the beginning of a sentence, we use a comma at the end of this dependent clause. However, it's not necessary to put a comma before since when this dependent clause starts after the main clause.
We are not referring to a particular emotion. Instead, we are referring to all type of emotions. So we must use "emotions".
I don't think there is anything unnatutal or non idiomatic in the sentence in question.
These are nearly synonymous phrases. Within is being used as a preposition to indicate that you are referencing the area inside of the radius in relation to the radius. In basically communicates the same message, but with less focus on the actual radius.
However, “within” and “in” are not always interchangeable. “Within” describes an object’s placement for location and time.
- The building is within the city limits
- You will receive your order within five business days
“Within” is specifically used to describe the boundaries of a location or time, consequently placing a larger emphasis on constraints. “Within six hours” implies that something will happen before six hours. However, “in six hours” suggests a more flexible time range (something will happen in about six hours).
“In” is used for a wider variety of situations, such as inclusion (there are many more), and is used in many idiomatic expressions as well.
- She serves in the military
- The delivery came just in time
Additionally, the word “meter” in your phrases should be plural because multiple meters are modifying the radius.
To answer the second part of your question, I believe “on the border/edge/circumference of the circle” most clearly communicates that you are referencing the border of the circle and not the area it encloses.
Another note: you can rewrite the phrases as “within/in a 10 meter radius” to make them more concise. This has no effect on the meaning.
Best Answer
The verb is linked to the noun "the computed X, Y and Z" where all three can be looked collectively. Hence they are plural and plural verb will be used. i.e. "were"