A single-lane highway must pass under a series of bridges....

Name: Problem 748, Chapter 42: Geometry
Uploaded: 2021-01-19T16:55:22-08:00
Duration: 3 min 39 s
Channel: Sheryl Ezze
Description: A single-lane highway must pass under a series of bridges. It is proposed that the bridges be shaped as semi-ellipses with the height equal to the width. The builder feels he must allow room for a 6 foot wide, 12 foot high truck to pass under it. What is the lowest bridge that can be built to serve this purpose.

A single-lane highway must pass under a series of bridges. It is proposed that the bridges be shaped as semi-ellipses with the height equal to the width. The builder feels he must allow room for a 6 foot wide, 12 foot high truck to pass under it. What is the lowest bridge that can be built to serve this purpose.

Key Concepts

Ellipse Equation

An ellipse is defined by the equation (x²/a²) + (y²/b²) = 1, where a and b are the lengths of the semi-major and semi-minor axes respectively. This equation provides a relationship between any point (x, y) on the ellipse and its parameters, and is fundamental when designing curved structures based on elliptical shapes.

Semi-Elliptical Arch

A semi-elliptical arch is formed by taking the upper (or lower) half of an ellipse. In such structural designs, the arch has a defined height and width which correlate with the ellipse’s axes, and it is frequently used in architecture and engineering for its aesthetic and structural properties.

Geometric Constraint Optimization

When a structure must meet specific clearance or dimensional requirements, geometric constraints are applied to the ellipse’s equation. By imposing conditions at particular points along the arch—such as ensuring the height at a given horizontal distance meets the required clearance—it is possible to solve for the necessary dimensions that optimize both safety and material use.

Transcript

00:01 So we have an arch that's the shape of an ellipse.

00:03 So let's kind of place it into a coordinate system.

00:07 And we know the arch is shaped like this.

00:10 And we know that it's supposed to be 20 feet across, which means this point would be the point 10.

00:18 And this point would be the point negative 10 .0.

00:22 And i said that the maximum height is 12 feet.

00:25 So this point would be 0 and 12 feet.

00:28 So we know that in this ellipse.

00:31 Lips that little a is equal to 10 and i'm sorry little a is equal to 12 it's actually elliptical that 12 feet is here we would have the little a is going to be this this is our little a and this is actually our little b which is kind of different b is 10 so we can write the equation of our ellipse and our ellipse is x squared over in this case because that's actually going to be long in this direction x squared over 10 squared plus y squared over and that will have the a underneath it because in the y direction this goes underneath the y values 144 equals 1 and our question is if we have this truck and we have this truck that has a load on it and we know that it is 10 feet wide and it is a 11 feet tall.

01:35 The question is is it going to fit underneath here? so what do we need to do? we need to have it go down the center and we need to find where this 5 feet is, where that 5 is.

01:47 And we need to know what is its height because it's going to have to go through like this and if that height is not we need that height and if that height is not at least 11 feet, we're going to have a problem...

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