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so this problem they're telling us that 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 it 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 3,000 miles in if asteroid travels 3,000 miles in six hours what is the constant of variation so the distance is 3000 so we have D is equal to 3,000 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 want to 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 506 divided by 6 is 1 the hours also cancel out if we care about the units and so the constant of proportionality the left hand side is just 500 500 and then we have miles per hour miles per hour 5 500 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 unit's or we should say the constant of variation to use the terminology that they actually use in the question