/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 81 Arc Length Use the integration c... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 10: Problem 81

Arc Length Use the integration capabilities of a graphing utility to approximate to two-decimal-place accuracy the elliptical integral representing the circumference of the ellipse $$ \frac{x^{2}}{25}+\frac{y^{2}}{49}=1 $$

Short Answer

Expert verified
The circumference of the ellipse is approximately four times the product of the semi-major axis and the complete elliptic integral of the second kind when evaluated with the value of the eccentricity.

Step by step solution

01

Identify a and b

In the given ellipse's equation \(\frac{x^{2}}{25}+\frac{y^{2}}{49}=1\), \(a^2\) and \(b^2\) are the denominators of the fractions. So, identify that \(a = 7\) and \(b = 5\).
02

Calculation of eccentricity (e)

The eccentricity of the ellipse is given by the formula \(e = \sqrt{1 - (b^2 / a^2)}\). So, substitute \(a = 7\) and \(b = 5\) into the formula to find \(e = \sqrt{1 - (5^2 / 7^2)} = \sqrt{1 - 25/49} = \sqrt{24/49} = 0.699 \) to three decimal places.
03

Use the elliptical integral formula

Now use a graphing utility to calculate the complete elliptic integral of the second kind using the calculated value of e. The exact detail of how to do this will depend on the specific graphing tool used, but typically this involves inputting the function E(e) into the tool and evaluating it.
04

Calculation of circumference

Now that E(e) is known, substitute \( a = 7 \) and the calculated E(e) back into the initial formula to find the circumference. \( L = 4aE(e) = 4*7*E(0.699) \).

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

Most popular questions from this chapter

Finding a Polar Equation In Exercises \(33-44\) , find a polar equation for the conic with its focus at the pole. (For convenience, the equation for the directrix is given in rectangular form.) Conic \(\quad\) Eccentricity \(\quad\) Directrix Ellipse \(\quad e=\frac{3}{4} \quad y=-2\)

Finding the Area of a Polar Region Between Two Curves In Exercises \(43-46,\) find the area of the region. Inside \(r=2 a \cos \theta\) and outside \(r=a\)

Finding the Area of a Polar Region In Exercises \(5-16\) , find the area of the region. Interior of \(r^{2}=6 \sin 2 \theta\)

Identifying a Conic In Exercises \(23-26,\) use a graphing utility to graph the polar equation. Identify the graph and find its eccentricity. $$ r=\frac{3}{-4+2 \sin \theta} $$

Sketching and Identifying a Conic In Exercises \(13-22\) , find the eccentricity and the distance from the pole to the directrix of the conic. Then sketch and identify the graph. Use a graphing utility to confirm your results. $$ r=\frac{6}{2+\cos \theta} $$
See all solutions

Recommended explanations on Math Textbooks

Applied Mathematics

Read Explanation

Calculus

Read Explanation

Probability and Statistics

Read Explanation

Mechanics Maths

Read Explanation

Logic and Functions

Read Explanation

Pure Maths

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.