Direct variation word problem: space travel

0:00So, 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.

0:14So, 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.

0:38The 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.

1:01We 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.

1:23And, 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.

1:48So, 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.