A new bridge is to be constructed over the east river in NewYork City. The space

Question

A new bridge is to be constructed over the east river in NewYork City. The space between the supports needs to be 1050feet; the height at the center of the arch needs to be 350feet. Two structural possibilities exist: the support couldbe in the shape of a parabola or the support could be in the shapeof a semi ellipse.      An empty tanker needs a 280-footclearance to pass beneath the bridge. The width of thechannel for each of the two plans must be determined to verify thatthe tanker can pass through the bridge. (a) Determine the equation of a parabola with thesecharacteristics. [Hint: Place the vertex of the parabola at theorigin to simplify calculations.] (b) How wide is the channel that the tanker can passthrough? (c) Determine the equation of a semi ellipse with thesecharacteristics. [Hint: Place the center of the semi ellipse at theorigin to simplify calculations.] (d) How wide is the channel that the tanker can passthrough? (e) If the river were to flood and rise 10 feet, how would theclearances of the two bridges be affected? Does this affectyour decision as to which design to choose? Why?

Explanation / Answer

a) the equation of a parabola is y=a(x-h)2+ k. as the question hints, the vertex of the parabola issituated at the point (0,0) to simplify the problem. since (0,0) isthe vertex of the parabola, h and k are zero. then the next thingto find would be the value of a. since the curve points downwards,a must be negative, the new equation looks like. y=-ax2.to find a, you would just plug in values for x and y. this isobtained from the picture. since the height of the parabola is 350feet, and the parabola points downwards, -350 would be your yvalue. and since the total width of the bridge is 1050 feethalf of it gives you your x value. therefore, you just need to plugin -350 and 525 into the y and x variables, respectively, of theequation and you should get your a value. plug the a value into theequation and you get your equation.

b) the width of the channel is the x value when you plug in theheight at that point into the equation you just got and solve fory. so in your equation, (y=-ax2) you plug in your yvalue and since you got your already, x is the only variable. solvefor x. you obtain your y value by subtracting the clearance needed(280 ft.) from the height of the parabola (350 ft.). you then plugit into the equation and solve for x.

c) the equation of an ellipse is((x-h)2/a2) +((y-k)2/b2) = 1. since the center of theellipse will be situated at the origin, h and k are zero. a is thewidth from the center to the farthest point out on the x-axis. thatvalue is given by half of the total width which is half of 1050 ft.b is given by the distance from the center to the tallest pointwhich is straight above it. this value is given by the heightneeded for the bridge which is 350 ft. by solving for y, you getthe equation of the ellipse. use the positive answer because it isabove the x-axis. (when you square root something you get plus orminus the value. that plus and minus gives you the whole parabola.you only need half of it.)

Related Questions

Navigate

• Browse (All)
• Browse A
• Subjects

---

---

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.