Example finding appropriate units

Louisa runs a lawn mowing business. She decides to measure the rate at which the volume of fuel she uses increases with the area of the lawn. What would be an appropriate unit for Louisa's purpose?

So let me reread this to make sure I understand it. She decides to measure; she's going to measure the rate at which the volume of fuel she uses increases with the area of the lawn. So her unit should be something that says volume per area. She wants to see how her volume of fuel increases with the area of the lawn, so it should be volume per area that should be her rate.

So let's think about which of these units give us volume per area.

So liters right over here, that is definitely volume, but this right over here isn't area; this is distance. If it said square kilometers or kilometers squared, then we'd be in business, so we could rule this one out.

Centimeters per kilometer, well this is a distance per distance, not a volume per area, so we can rule that out. Centimeters per square kilometers, well this is a distance per area or distance per distance squared, but that is not volume per area, so let's rule that one out.

Now, liters per kilometer squared, this one's looking good. This is a volume in liters, and this right over here is an area. This isn't the only one that would have worked, but this is out of the choices, the only one that works. If they had given something like a pint per foot squared or a pint per square foot or even if it was meter cubed per kilometers squared, that would have also been volume per area, but this is the best choice of these.

Let's do another one of these. Snow is piling outside Cameron's house. He decides to measure the rate at which the height of the pile increases over time. What would be an appropriate unit for Cameron's purposes? Like always, pause the video and see if you can figure it out on your own.

So the rate at which height increases over time should be a height per time, or really I should say a distance per time is what the height is going to be doing. The rate at which the height is changing over time; it should be a distance or maybe even better, a length—length per unit time.

So let's see; this is hours per meter, so this is time per length. This is the reciprocal of that. This one right over here is time per length, as opposed to length per time, so I would rule that one out.

Liters per minute; this is a volume per unit time, not a length per unit time—rule that one out. Minutes per liter; what is that? It's time per volume. Well, we don't want to do a time per volume; we want a length per time.

Meters, that's a length per time, hour—that one works. Meters per hour; this is a length, and that is a time. And it's good to always go back to the original context—he decides to measure the rate at which the height of the pile increases over time.

So if someone said, "Hey, that height of that pile is increasing five meters," that would be too much for snow. But let's say it's increasing half a meter per hour, which even that would be quite fast. But half a meter per hour—that makes sense in your brain.

Hey, every hour I'm going to get half a meter more of snow. If someone were to tell you, "Hey, the snow outside is increasing at a rate of five liters per minute," well that right there—could be maybe the volume of snow, maybe over your entire lawn or something, but that would not be giving you the height per time. This is height per time; it would be length per time, right over here.