Tony E. Wong, *Rochester Institute of Technology*

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If everyone on Earth sat down in the ocean, how far would the water rise? - Zahkaev and Viktor

Hypothetical questions, like what would happen if everyone on Earth went for an ocean swim at once, are fun to think about. And using math, you can get pretty close to a real answer. Let's start by considering a smaller version of the same question.

## Bathtub math

If you fill a bathtub all the way to the top and hop in, you know you're in for a soggy cleanup. The water overflows because your body pushes it out of the way - something called displacement. Since the tub has a solid bottom and sides, the only direction the water can go is up and out.

The amount of space an object - in this case, you - takes up is called volume. The volume of water that overflows the tub is equal to the volume of your body.

Now think about a situation where the bathtub is only half full. As you hop in, the volume of your body still pushes the water up. You can calculate how much the water level in the tub will rise with a few simple math equations.

Suppose the bathtub is a rectangular box. You can figure out how much the water level will rise when you sit down in the tub by considering how much volume you are adding to the tub and what size area you are spreading this volume over. The amount the water level rises is equal to the added volume divided by the area.

For a bathtub that is 5 feet long and 2 feet wide, the area is 10 square feet.

Now, let's figure out your volume. To make the math easier, let's suppose that you, like the bathtub, are also a rectangular box. Let's say you are about 4 feet tall and 2 feet wide (from left to right) and 1 foot deep (from front to back). The volume of your body would be 4 feet x 2 feet x 1 foot, or 8 cubic feet.

When you sit down, you are adding the volume of approximately half your body to the tub. This means the height of the water level rise is equal to the volume of half your body, divided by the area of the tub. Using the estimates above, this leads to a water level rise of 4 cubic feet divided by 10 square feet, which equals about 5 inches. You would certainly notice that!

## Scaling up

You can think about the oceans as a gigantic bathtub. More than 70% of the Earth's surface is ocean, giving this bathtub an area of about 140 million square miles. To figure out how much the water will rise, we need to know the volume of people sitting in it and divide it by this ocean area.

Currently, there are almost 8 billion people on Earth. Human beings come in all sizes, from tiny babies to large adults. Let's assume the average size is 5 feet tall - a bit bigger than a child - with an average volume of 10 cubic feet. Only half of each person's body would be submerged when they sit down, so only 5 cubic feet adds to the water level. With 8 billion people total, you can calculate 5 x 8 billion which gives a whopping 40 billion cubic feet that would be added to the oceans.

But remember, this volume would be spread over the vast area of the oceans. Using the same bathtub math as before, we divide the 40 billion cubic feet of volume over the 140 million square miles of ocean.

The answer? The total rise in sea level would be about 0.00012 of an inch, or less than 1/1000th of an inch. If everyone completely submerged themselves, this would double the answer to 0.00024 inches, which is still only about the width of a human hair.

It turns out the oceans are enormous - and humans are just a drop in the bucket.

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Tony E. Wong, Assistant Professor of Mathematical Sciences, *Rochester Institute of Technology*

This article is republished from The Conversation under a Creative Commons license. Read the original article.