Let us call the time as [tex] t [/tex] and the distance as [tex] d [/tex].
It is given that the time between seeing lightning and hearing thunder varies directly with the distance away from the lightning.
Mathematically, this direct variation is represented as:
[tex] t\propto d [/tex]
Now, to remove the [tex] \propto [/tex] sign we replace it by an equal to sign (=) and a constant of variation, k. (Please note that this is an important step)
Thus, the above equation becomes:
[tex] t=k\times d [/tex]
In order to find the constant of variation, k, we isolate k by dividing both sides of the above equation by d. Thus, we get:
[tex] \therefore k=\frac{t}{d} [/tex]
Plugging in the values of t and d given in the question, we get:
[tex] k=\frac{10 seconds}{2 miles} =5 [/tex] seconds per mile.
Therefore, the constant of variation for this problem is 5 seconds per mile.