10. Find the area of each of the following. Leave your answers in terms of . (a) 5 cm 3 cm 60° (b) (d) 4 cm 20 cm 0 10 cm
10. Find the area of each of the following. Leave your answers in terms of . (a) 5 cm 3 cm 60° (b) (d) 4 cm 20 cm 0 10 cm
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
![**Problem 10: Calculation of Areas**
Find the area of each of the following. Leave your answers in terms of \( \pi \).
- **Diagram (a): Circle**
- A circle with a radius of 5 cm.
- To find the area, use the formula:
\[
\text{Area} = \pi r^2
\]
- Substituting the given radius:
\[
\text{Area} = \pi (5)^2 = 25\pi
\]
- **Diagram (b): Sector**
- A sector with a central angle of \(60^\circ\) and a radius of 4 cm.
- To find the area of the sector, use the formula:
\[
\text{Area} = \frac{\theta}{360^\circ} \times \pi r^2
\]
- Substituting the given values:
\[
\text{Area} = \frac{60}{360} \times \pi (4)^2 = \frac{1}{6} \times 16\pi = \frac{8\pi}{3}
\]
- **Diagram (c): Semicircle**
- A semicircle with a radius of 3 cm.
- To find the area of the semicircle, use the formula:
\[
\text{Area} = \frac{1}{2} \times \pi r^2
\]
- Substituting the given radius:
\[
\text{Area} = \frac{1}{2} \times \pi (3)^2 = \frac{9\pi}{2}
\]
- **Diagram (d): Annular Sector**
- Two concentric circles forming an annular sector with a central angle \( \theta \). The outer radius is 20 cm, and the inner radius is 10 cm.
- To find the area of the annular sector, calculate the area of the larger sector and subtract the area of the smaller sector. Use the formula for a sector:
\[
\text{Area} = \frac{\theta}{360^\circ} \times \pi R^2 - \frac{\theta}{360^\circ} \times \pi r^2
\]
- Since \( \](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F79d4d229-649f-4192-a00b-385a235528b8%2F3db49033-f00b-45b1-ba78-95e2d7b9470e%2Fpfxwlx_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem 10: Calculation of Areas**
Find the area of each of the following. Leave your answers in terms of \( \pi \).
- **Diagram (a): Circle**
- A circle with a radius of 5 cm.
- To find the area, use the formula:
\[
\text{Area} = \pi r^2
\]
- Substituting the given radius:
\[
\text{Area} = \pi (5)^2 = 25\pi
\]
- **Diagram (b): Sector**
- A sector with a central angle of \(60^\circ\) and a radius of 4 cm.
- To find the area of the sector, use the formula:
\[
\text{Area} = \frac{\theta}{360^\circ} \times \pi r^2
\]
- Substituting the given values:
\[
\text{Area} = \frac{60}{360} \times \pi (4)^2 = \frac{1}{6} \times 16\pi = \frac{8\pi}{3}
\]
- **Diagram (c): Semicircle**
- A semicircle with a radius of 3 cm.
- To find the area of the semicircle, use the formula:
\[
\text{Area} = \frac{1}{2} \times \pi r^2
\]
- Substituting the given radius:
\[
\text{Area} = \frac{1}{2} \times \pi (3)^2 = \frac{9\pi}{2}
\]
- **Diagram (d): Annular Sector**
- Two concentric circles forming an annular sector with a central angle \( \theta \). The outer radius is 20 cm, and the inner radius is 10 cm.
- To find the area of the annular sector, calculate the area of the larger sector and subtract the area of the smaller sector. Use the formula for a sector:
\[
\text{Area} = \frac{\theta}{360^\circ} \times \pi R^2 - \frac{\theta}{360^\circ} \times \pi r^2
\]
- Since \( \
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
Step by step
Solved in 3 steps with 3 images
Recommended textbooks for you
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
