Math in the News

Issue 98

THE SEARCH FOR MISSING FLIGHT MH370

• Malaysian Airlines Flight 370 was a passenger flight scheduled to travel from Kuala Lumpur International Airport in Malaysia to Beijing Capital International Airport on March 8, 2014.

• The flight was last heard from March 8 at 1:20 Malaysian time.

• The aircraft was carrying 12 crew members and 227 passengers from 14 nations.

THE SEARCH FOR FLIGHT MH370

The intended flight path was direct, expected to fly Northeast until it landed in Beijing.

THE SEARCH FOR FLIGHT MH370

However, information from

Malaysian military satellites indicate

that the flight veered West shortly after

takeoff.

THE SEARCH FOR FLIGHT MH370

THE SEARCH FOR FLIGHT MH370

Select the image to watch a BBC video on the flight path and the search for MH370

http://www.bbc.com/news/world-asia-26503141

Authorities confirmed that the airplane crashed somewhere in the Indian Ocean and have begun their search in the vast sea. Air satellites have detected some debris in the water and satellite “pingers” have been dropped into the ocean to detect signals from the plane’s black box.

THE SEARCH FOR FLIGHT MH370

Search area for flight MH370

Based on the location of the pings and debris fields, mathematicians are attempting to narrow down the search area. Coordinate geometry allows us to map the search area.

THE SEARCH FOR FLIGHT MH370

French satellite imagery showing potential debris on March, 23, 2014

Coordinate geometry maps coordinates on a coordinate plane, such as the one shown on the right. The plan has a horizontal axis (x) and a vertical axis (y). Coordinates are written in (x,y) format where the x indicates left/right movement and the y indicates up/down movement.

THE SEARCH FOR FLIGHT MH370

x

y

(15,10)

If we have enough coordinates, we can map the area. This is the method authorities are using to find MH370. Let’s try an example.

THE SEARCH FOR #MH370

A

DC

B

The Surveyor’s Area Formula, also known as the Shoelace Formula, can be used to find the area of any polygon on a grid. Those looking for flight MH370 will use this technique to determine the search area. The following shows the area for a quadrilateral, but the formula can be used for any polygon.

THE SEARCH FOR FLIGHT MH370

Continue to see how to use the formula and how it got its name…

The Surveyor’s Formula was first used for land surveyors to estimate the an area of land using coordinates. It is derived using determinants corresponding to the vectors of the edges of a polygon.

Determinants =

Based on determinants, the formula is written as

After the determinants are multiplied out, the formula can be re-written for quadrilaterals as:

THE SEARCH FOR FLIGHT MH370

To use the Surveyor’s Formula, we first need to lay out the coordinates in a table as they sit counterclockwise

THE SEARCH FOR FLIGHT MH370

Coordinate (x, y)

A (15,10)

B (5, 15)

C (-10, -10)

D (20, -10)

THE SEARCH FOR FLIGHT MH370

Coordinate x y

A 15 10

B 5 15

C -10 -10

D 20 -10

A 15 10 The formula got its second name, the Shoelace formula, because of how it looks when you cross multiply!

Now, we can plug the points into the formula. First, let’s do the math inside the absolute value sign…

(15 • 15) + (5 • -10) + (-10 • -10) + (20 • 10)– (5 • 10) – (-10 • 15) – (20 • -10) – (15 • -10) =

(225) + (-50) + (100) + (200)– (50) – (-150) – (-200) – (-150) = 925

Now, we can finish the equation…

THE SEARCH FOR FLIGHT MH370

The area of the polygon on the coordinate plane is 462.5 square units.

1/2│925│= 462.5

With the same method, you can use the coordinates of the debris to find one of the potential search areas. The world is mapped much like a coordinate plane, where East and West represent the x-axis and North and South represent the y-axis.

THE SEARCH FOR FLIGHT MH370

• Consider the following questions:

• In what units is the search area measured?

• How does the depth of the ocean impact the difficulty of the search?

• Take a look at this video update on the search for flight MH370. How does this impact the search? Can we still use the Surveyor’s Formula with the new search method?

THE SEARCH FOR FLIGHT MH370