Book edition 6th

Author(s) Sullivan

Pages 1200 pages

ISBN 9780321795465

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Short Answer

Semi elliptical Arch Bridge The arch of a bridge is a semi ellipse with a horizontal major axis. The span is $30$ feet, and the top of the arch is $10$ feet above the major axis. The roadway is horizontal and is $2$ feet above the top of the arch. Find the vertical distance from the roadway to the arch at $5$ -foot intervals along the roadway.

The vertical distance from the roadway to arch at $5$ foot intervals are $2.57$ feet , $4.54$ feet and $12$ feet.

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Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1. Given information.

The major axis of the arch is parallel to the x-axis. since the span of the arch is $30$ feet, this means

$a = \frac{30}{2} a = 15$

The top of the arch is $10$ feet above the major axis.This means $b = 10$ .

The road ways lies $2$ feet above top of the arch.Thus, the coordinates of the center are $(0, - 12)$ .

Step 2. The equation of the ellipse and graph of the arch.

The equation of the ellipse is $\frac{{(x - 0)}^{2}}{225} + \frac{{(y + 12)}^{2}}{100} = 1$

The graph of the arch is

Step 3. Vertical distance from roadway to arch.

Now, putting $x = 5$ in the equation of ellipse

role="math" localid="1646806151750"

\frac{25}{225} + \frac{{(y + 12)}^{2}}{100} = 1 {(y + 12)}^{2} = \frac{800}{9} y = \pm \frac{20 \sqrt{2}}{3} - 12 y = - 2.57, - 21.42

We have considered only the upper semi- ellipse constituting the arch.

Putting $x = 10$ in the equation we get.

role="math" localid="1646806607812" $\frac{100}{225} + \frac{{(y + 12)}^{2}}{100} = 1 {(y + 12)}^{2} = \frac{500}{9} y = \pm \frac{10 \sqrt{5}}{3} - 12 y = - 4.54$

For $x = - 15$ and $x = 15$ $y = - 12$

Thus, the vertical distances from the roadway to arch at $5$ foot intervals are $2.57 f e e t, 4.54 f e e t$ and $12 f e e t$ .

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Precalculus Enhanced with Graphing Utilities

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Short Answer

Step by Step Solution

Step 1. Given information.

Step 2. The equation of the ellipse and graph of the arch.

Step 3. Vertical distance from roadway to arch.

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Precalculus Enhanced with Graphing Utilities

Answers without the blur.

Short Answer

Step by Step Solution

Step 1. Given information.

Step 2. The equation of the ellipse and graph of the arch.

Step 3. Vertical distance from roadway to arch.

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