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So, in this problem, they're telling us in
outer space, the distance an object travels varies directly with the
amount of time that it travels. And, that's of course assuming that it's
not accelerating, and there's no net force, and all of that on
it. So, I guess they're talking about a
specific object. So, some specific object, the amount, the
distance that it travels is directly, it varies directly, it varies directly
with the amount of time that it travels. So, if we think of in, in, in terms of constants of proportionality, and direct
variation, we could say that the distance, we could say that the distance
is equal to some constant, times the time, times the
time that it travels. The distance varies directly with the
amount of time that travels for this particular
object. If an asteroid travels 3000 miles, in, if
the asteroid travels 3000 miles in 6 hours, what is the
constant of variation? So the distance is 3000, so we have d is
equal to 3000 miles. We have 3,000 miles is the distance, and
that's going to be equal to the constant of
variation. The constant of variation times the time,
times 6 hours. So, if we wanna solve for the constant of variation, we can just divide both sides
by 6 hours. 6 hours, and we divide the right-hand side
by 6 hours. And, so, 3,000 divided by 6 is 500, and 6
divided by 6 is 1. The hours also cancel out if you care
about the units. And, so, the concept of proportionality,
the left-hand side is just 500, 500, and then we have miles per hour,
miles per hour. Fired 500 miles per hour, and that is
equal to k. So, the constant proportionality is 500
miles per hour, or you could say 500 if you're not too worried
about the units. Or, we should say, the constant of
variation, to use the terminology that they actually use
in the question.