3580.14 – Akashi-Kaikyo Bridge

Pictured below is a schematic drawing of a suspension bridge. The longest single span suspension bridge in the world is the Akashi-Kaikyo Bridge in Japan. Google it – it's an engineering marvel. The span at sea level ( $CD$ ) is 1991 meters (more than a mile). The tops of the towers are 282.85 meters above the water ( $AC$ = $BD$ ), and are 89 centimeters farther apart than at the water level.

So? What's the radius of the earth?

Solution

The radius of the earth is $x$ kilometers. Being careful with the decimal points, we work in kilometers, since that's the best unit to use for the radius of the earth.

$1991 \ \text{m} = 1.991 \ \text{km}$ , $282.85 \ \text{m} = 0.28285 \ \text{km}$ , and $89 \ \text{cm} = 0.000089 \ \text{km}$ .

From the picture below (not to scale!), we derive the proportion $\dfrac{x}{1.991}=\dfrac{x + 0.28285}{1.991 + 0.000089}$ .

Then $x(1.991+0.000089)=1.991(x+0.28285)$ .

On we go: $1.991089x = 1.991x + 0.28285 \leadsto 0.000089x = 0.28285 \leadsto x = \dfrac{0.28285}{0.000089} = 6327 \ \text{km}$ for the radius.

Really? Well,since the earth is not a perfect sphere, its radius varies between 6357 km and 6378 km. Given the accuracy of the measurements given in the problem, our answer of 6327 km is pretty good!

Extension of the problem: Take a radius of the earth that's within the interval given, say, 6360 km, and work backwards to find out what the distance between the tops of the towers would be. Better yet, find out what the radius of the earth is at the place where the bridge is, and use that figure.

(The span of the Golden Gate Bridge is 1280 meters.)