![]() What about curves in three or more dimensions? One common exercise in a standard calculus course is to find the arc length of a helix. When possible, Wolfram|Alpha returns an exact answer in this case the answer involves the hyperbolic sine function, which you can then have Wolfram|Alpha approximate to any desired accuracy using the More digits button on the right. Notice that Wolfram|Alpha shows the calculation needed to find the arc length (just like finding an area under a curve, integration is required) as well as the answer. So the main part of each cable is about 4,354 feet long-slightly more than the distance between the towers. From this information we can use Wolfram|Alpha to find the equation defining the parabolic curve of the cables:Įntering the equation from the “Equation forms” pod into the input box, we now ask Wolfram|Alpha for the length of each cable over the main span: The Golden Gate Bridge, shown above, has a main span of 4,200 feet and two main cables that hang down 500 feet from the top of each tower to the roadway in the middle. The shape in which a cable hangs by itself is called a “catenary,” but with a flat weight like a roadway hanging from it, it takes the shape of a more familiar curve: a parabola. To see why this is useful, think of how much cable you would need to hang a suspension bridge. You can also think of it as the distance you would travel if you went from one point to another along a curve, rather than directly along a straight line between the points. An arc length is the length of the curve if it were “rectified,” or pulled out into a straight line. One of the features of calculus is the ability to determine the arc length or surface area of a curve or surface.
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