Q 70. Semi elliptical Arch Bridge The ... [FREE SOLUTION]

91影视

Precalculus Enhanced with Graphing Utilities

Sullivan

$Math Studyset 91影视 Explanations$ Math

6th Edition 路 1200 pages

ISBN: 9780321795465

Chapter 10: Q 70. (page 655)

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.

Short Answer

Expert verified

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

Step by step solution

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

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 $12 f e e t$ .

Unlock Step-by-Step Solutions & Ace Your Exams!

Full Textbook Solutions
Get detailed explanations and key concepts
Unlimited Al creation
Al flashcards, explanations, exams and more...
Ads-free access
To over 500 millions flashcards
Money-back guarantee
We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Recommended explanations on Math Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

91影视

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.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Mechanics Maths

Pure Maths

Applied Mathematics

Probability and Statistics

Statistics

Theoretical and Mathematical Physics

Study anywhere. Anytime. Across all devices.