Chapter 6: Q. 78 (page 540)

Use Theorem 6.7 to prove that a circle of radius 5 has circumference $10 π$ .

Short Answer

Expert verified

Circle of radius 5 has circumference $10 π$ .

Step by step solution

Step 1. Given Information

Circle of radius 5.

Step 2. Proving circumference of circle of radius 5 to be 10π.

$Then the arc length of f (x) from x = a to x = b can be represented by the definite integral : \int_{a}^{b} \sqrt{1 + (f' {(x))}^{2}} d x Equation of circle of radius 5 i s : x^{2} + y^{2} = 5^{2} Therefore, y = \sqrt{5^{2} - x^{2}} = f (x) and f' (x) = - \frac{x}{\sqrt{5^{2} - x^{2}}} Circumference of the circle is twice the arc length from x = - 5 to x = 5 of f (x) Therefore, C = 2 \int_{- 5}^{5} \sqrt{1 + (f' {(x))}^{2}} d x C = 2 \int_{- 5}^{5} \sqrt{1 + {(- \frac{x}{\sqrt{5^{2} - x^{2}}})}^{2}} d x C = 2 \int_{- 5}^{5} \sqrt{1 + (\frac{x^{2}}{5^{2} - x^{2}})} d x C = 2 \int_{- 5}^{5} \sqrt{{\frac{5^{2} - x^{2} + x}{5^{2} - x^{2}}}^{2}} d x C = 2 \int_{- 5}^{5} \frac{5}{\sqrt{5^{2} - x^{2}}} d x Put, x = 5 \sin t, dx = 5 \cos t dt Limits of integration change to - \frac{π}{2} to \frac{π}{2} C = 2 \int_{- \frac{π}{2}}^{\frac{π}{2}} \frac{5 \times 5 \times \cos t}{\sqrt{5^{2} - {(5 \sin t)}^{2}}} dt C = 2 \int_{- \frac{π}{2}}^{\frac{π}{2}} \frac{5 \times 5 \times \cos t}{\sqrt{5^{2} - {(5 \sin t)}^{2}}} dt C = 10 \int_{- \frac{π}{2}}^{\frac{π}{2}} \frac{5 \times \cos t}{5 \times \cos t} dt C = 10 \int_{- \frac{π}{2}}^{\frac{π}{2}} 1 dt C = 10 t_{- \frac{π}{2}}^{\frac{π}{2}} C = 10 [\frac{π}{2} - (- \frac{π}{2})] C = 10 π$

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.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

Recommended explanations on Math Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Use Theorem 6.7 to prove that a circle of radius 5 has circumference $10 π$ .

Short Answer

Step by step solution

Step 1. Given Information

Step 2. Proving circumference of circle of radius 5 to be 10π.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Pure Maths

Decision Maths

Statistics

Logic and Functions

Geometry

Calculus

Study anywhere. Anytime. Across all devices.

Use Theorem 6.7 to prove that a circle of radius 5 has circumference10π.

Short Answer

Step by step solution

Step 1. Given Information

Step 2. Proving circumference of circle of radius 5 to be 10π.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Pure Maths

Decision Maths

Statistics

Logic and Functions

Geometry

Calculus

Study anywhere. Anytime. Across all devices.

Use Theorem 6.7 to prove that a circle of radius 5 has circumference $10 π$ .