Pre Calculus II Project Setup:

Pre Calculus II Project

Setup: the x-axis as the water level. The origin is the midpoint between the two x-intercepts/support bases. The clearance level is a horizontal line at QUOTE. Three points are given that are the same for both parabola and semiellipse: the first is the y-intercept, which occurs at 350' above the water, or QUOTE; the two other points are the x-intercepts. The distance between the two supports is 1050', and since the y-intercept bisects both shapes, the x-intercepts are at QUOTE and QUOTE. Therefore, the three points are: QUOTE, QUOTE, and QUOTE.

The equation for the parabola can be obtained using those three points. The parabola is centered, so the x-term is not added to or subtracted from before being multiplied by its coefficient. The parabola must also be negative to create the necessary shape. Using the y-intercept, we can create the preliminary equation QUOTE, where a is a coefficient that must be solved for. To do this, we use an x-intercept. Thus:

To find the width of the channel through which the tanker can pass, one must find the length of the value of x when the parabola is intersected by the horizontal line QUOTE. Hence:

Hence, the parabolic bridge allows 234.287 ft of clearance from the centerline, or about 470 ft in all.

The equation for the ellipse is found in similar fashion. The semi-minor and semi-major axis are easily determined, and can then be subbed into the standard equation for an ellipse. Taking the square root of y will result in a plus/minus, and discarding the minus erases the lower half of the ellipse. The long axis extends horizontally, and the short axis extends vertically. The x and y axis bisects the ellipse already, so both a and B. are available: 525' and 350'.

The width of the channel is, once again, determined by inputting a known point.

Hence, the semi-elliptic bridge allows 315 ft of clearance from the centerline, or 630 ft in all.

If the river flooded, the tanker would sit 10 ft higher and would, hence, have only the clearance available at 290 ft. For the parabola:

For the semi-ellipse: