# Finding Flight 370: The Math and the Meteorology

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Flight 370 (a Boeing 777) has disappeared – could math and meteorology help find the missing plane? Photo credit M42

Malaysia Flight 370, a Boeing 777, has disappeared. This plane is a 300 ton object that easily blots out the sun if one stands underneath a wing.

How long should it take to find such an aircraft in the air, on the ground, or in the ocean? That depends mostly on the amount of territory that has to be searched.

## Flight 370: How the Atmosphere Affects the Range of the Plane

An airplane needs a sufficient density of the air flowing over the wings in order to maintain stability. The density of air above about 8 miles in altitude falls below the critical value for a Boeing 777. If the plane flies too high, it will become uncontrollable and crash, literally vanishing into thin air.

## The Volume of Space Below Eight Miles

The surface area of a sphere with radius r is A=4πr^2. The radius of the earth is about 4,000 miles, so its surface area is approximately 200,000,000 square miles. Thus the volume of space below eight miles is this number multiplied by eight — about one and one-half billion cubic miles. If an aircraft had an infinite amount of fuel, it could fly around undetected for a very long time. Even given that radar covers most land areas, a plane continuing to fly over oceans would have a billion cubic miles in which to disappear.

## The Range of a Boeing 777

Flight 370 was headed from Kuala Lumpur to Beijing, a distance of 2700 miles. Though the Boeing 777 has a range of 5200 miles on a full tank of fuel, commercial airliners rarely carry more than enough fuel to take them to their destination and a few hundred additional miles. The range of Malaysia 370 was certainly not more than 3500 miles. Furthermore, it is nearly impossible for the pilots of this plane to have found a place at which to refuel and take off again undetected.

Because the aircraft is no longer in the air, the search can be confined to the land and sea surface within a radius of 3500 miles from Kuala Lumpur. A circle with a radius of 3500 miles has an area A=πr^2, which equals 40 million square miles. The curvature of the earth introduces no more than a rounding error.

It is clear from the flight path of the plane that the pilots wished to stay out of the range of ground-based radar; so we can take the area of the Indian Ocean, 28 million square miles, to be the area that investigators need to search.

## How Long Will It Take To Find The Plane?

A pilot in a small plane that flies 100 miles per hour with a visual range of two miles in every direction could search about three thousand square miles of the ocean’s surface in an eight-hour day. If 30 such planes were used, the debris of Flight 370 would be located in 30 days or less. Hopefully electronic signals picked up by satellite, satellite photography, and data from ships will narrow the search area. Provided there is visual evidence of the crash, reasonable assumptions lead to the conclusion that the wreckage should be found within a week.

## A Decidedly Gloomy Scenario

Much of the Indian Ocean is over ten  thousand feet deep. It is infinitely harder to find an object on the bottom of the deep ocean than at the surface of the land or sea. In 2009, Air France Flight 447 went down in ten thousand feet of water in the Atlantic Ocean. It took two years for investigators to recover the plane, despite the fact that the point of impact was known. If Malaysia 370 crashed in the deep Indian Ocean and sank without leaving any evidence at the surface, the plane may never be found.

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