A study of the Brooklyn Bridge Suspension Bridges A study of the Brooklyn Bridge

Background This activity would be appropriate for student at the 10th grade level The activity would proceed as follows: 1. Find out background about the Brooklyn Bridge/Suspension bridges in general 2. Discuss the parabolic shape of the main cable of suspension bridges 3. Use dynamic technology to “eyeball” a parabola that resembles the main cable of the Brooklyn Bridge (TI-nSPIRE) 4. Research the dimensions of the Brooklyn Bridge 5. Determine the equation of the parabola for the main cable of the bridge using TI-nSPIRE 6. Determine the cost of the cable using the equation

1. Facts about suspension bridges Suspension bridges are able to achieve longer spans than many other types of bridges They can be cheaper to build than other types of bridges In the NYC area, there are several suspension bridges, including the Brooklyn Bridge, Manhattan Bridge, Verrazano Bridge, George Washington Bridge, and others. Source: http://en.wikipedia.org/wiki/Suspension_bridge

2. The main cable of a suspension bridge is in the shape of a parabola. If a string (or cable) is suspended the shape formed is called a “catenary.” When the suspended cable is secured to the span of the bridge, it takes on a parabolic shape. Source:http://www.carondelet.pvt.k12.ca.us/Family/Math/03210/page5.htm

3. Use the dynamic “grab & move” feature of the TI-nSPIRE The basic parabola y=x2 does not look like the center cable of a suspension bridge

3. The “grab & move” feature allows us to stretch out the parabola This graph still does not visually resemble the center cable of a suspension bridge

These graphs look more as we expect a suspension bridge to appear.

4. What are the dimensions of the Brooklyn Bridge? Span between towers: 1596 feet Height of towers: 276.5 feet Height of towers above roadway: 117 feet Therefore, if we consider the lowest point to be the origin of a coordinate axis system, the parabola would intersect the points (-798,117), (0,0) and (798,117). [798=1596/2] Source:http://www.endex.com/gf/buildings/bbridge/bbridgefacts.htm and http://www.endex.com/gf/buildings/bbridge/bbgallery/bbroyal1883.htm

5. We can determine the equation for the parabolic cable We can fit a curve to the points(-798,117), (0,0) and (798,117)

5. Equation of the parabola The equation of the parabola according to the quadratic regression is y=.0001837x2 This can also be calculated by hand using algebra

The graph must be scaled properly It is impossible to see using the standard scale of the TI-nSPIRE

The zoom feature allows a better view

The equation of the parabola can be used to: Calculate the amount of cable necessary to build the bridge Calculate the cost of the cable Build a model of the bridge

Difficulties Students must know proper scaling when dealing with real life situations and large numbers Although the new TI-nSPIRE has a wide variety of capabilities, it is not user friendly The “regression” feature would seem mysterious to students; it would be ideal to do an algebraic derivation of this work and compare results to what the calculator found.

Extensions Students could compare the equation of the parabolic cable of the Brooklyn Bridge to the equation of other local suspension bridges Students could do a historical study of the Brooklyn Bridge

Assessment This type of project would require ongoing assessment. A rubric might include such categories as Uses a sensible coordinate axis system to represent graphs Is able to interpret results of the graph and apply to solving problems Is able to determine whether results make sense in the context of examination of the bridge