If all the people in the world jumped into Crater Lake, how much would the water level rise?


Crater Lake National Park

I have highlighted below some important information that can be used to solve this problem along with a few assumptions.

In March 2020, the world's population was estimated to be 7,800,000,000 people.   It took over 2 million years of human history for the world's population to reach 1 billion, and only 200 years more to reach 7 billion.   Dan Brown addresses population growth in his novel, Inferno.

According to Wikipedia, Crater Lake is 1,943 feet deep and has a surface area of 20.6 square miles.   Assume that the lake is roughly conical.

According to the website www.livescience.com, the average body mass, globally, was 136 pounds (62 kilograms).   In North America, which has the highest average body mass of any continent, the number was 178 pounds (80.7 kg). The density of water is approximately 62 pounds per cubic foot, and we are almost the same density, so the volume of the average human in the world is about 2.176 cubic feet.

When solving the problem, assume that the surface area of the Lake stays the same, so that only the height is affected by all the people jumping into the lake.   In reality, the surface area would probably expand, so the height would actually be smaller than the number you calculate.


Mrs. P at Crater Lake National Park in 2010

Extra Credit:
If all the people in the world junped into Lake Tahoe instead, how much would the water level rise?
According to Wikipedia, Lake Tahoe is 1,645 feet deep and has a surface area of 191.6 square miles.



Solution to the Problem:

There are two answers to this problem, depending upon your interpretation.
(1) If you used a cone for the new figure (after all the people jumped into the lake),
The water level in Crater Lake would rise 88.6 feet and the water level in Lake Tahoe would rise 9.5 feet.

(2) If you used a cylinder for the new figure (after all the people jumped into the lake),
The water level in Crater Lake would rise 29.5541469416138 feet and the water level in Lake Tahoe would rise 3.17753354382696 feet.

Here is the solution for (1) using a cone:
You must use the formula for the volume of a cone:


Convert square miles to square feet:
20.6 square miles = 20.6 (5280) (5280) = 574,295,040 square feet.

The volume of the water in Crater Lake = (1/3) (1943) (574,295,040) = 371,951,754,240 cubic feet.

When all 7.8 billion people jump in the lake, the volume will be 371,951,754,240 + 16,972,800,000 = 388,954,554,200 cubic feet.
The new depth would be 2031.66 feet
so the water level would rise by 88.6 feet.

Using similar calculations for Lake Tahoe:
Convert square miles to square feet:
191.6 square miles = 5,341,501,440 square feet.

The volume of the water in Lake Tahoe = (1/3) (1645) (5,341,501,440) = 2,928,923,289,600 cubic feet.

When all 7.8 billion people jump in the lake, the volume will be 2,928,923,289,600 + 16,972,800,000 = 2,945,896,089,600 cubic feet.
The new depth would be 1654.53 feet
so the water level would rise by 9.53 feet.

Here is the solution for (2) using a cylinder:
Colin Bowey, Ivy Joseph, and Kelly Stubblefield interpreted the words, "assume that the surface area of the Lake stays the same" meant that it would form a cylinder above the original cone.   Here are Colin's excellent solutions:

Crater Lake
Lake Tahoe

Colin also sent in solutions using a cone for the new figure and also a frustum of a cone.
Click here for his solution using a cone for the new figure


Correctly solved by:

1. Colin (Yowie) Bowey ** Beechworth, Victoria, Australia
2. Veena Mg Bangalore, Karnataka, India
3. Kelly Stubblefield ** Mobile, Alabama
4. Ivy Joseph ** Mumbai, Maharashtra, India

** solved the extra credit