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3 23/dost 100m 126 p² The diagram is a sector of a Circular Grdbord [Pal =10am and

3 23/dost 100m 126 p² The diagram is a sector of a Circular Grdbord [Pal =10am and

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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### Diagram Analysis and Problem The diagram represents a sector of a circular cardboard, with the following properties: - The length of the sides \( \overline{PQ} \) and \( \overline{PR} \) is 10 cm. - The angle \( \angle PQR \) is 126°. This sector is to be folded to form a cone such that \( \overline{PQ} \) and \( \overline{PR} \) coincide. #### Questions a) Calculate the surface area of the cone. b) Calculate the volume of the cone. - **Note**: Use \( \pi = \frac{22}{7} \). ### Graphical Explanation The diagram is a sector of a circle with: - Two straight lines \( \overline{PQ} \) and \( \overline{PR} \) of length 10 cm. - The included angle between these lines \( \angle PQR \) is 126°. When this sector is folded to form a cone: - The radius of the base of the cone can be derived from the arc length of the sector. - The slant height (10 cm) remains constant. ##### Computation: 1. **Arc Length Calculation**: - The arc length of the sector is part of the circumference of the circle from which it was cut. Using the formula for arc length, \( L = r \theta \) (where \( \theta \) is in radians), we calculate the length of the arc. 2. **Radius of Cone's Base**: - The arc length equals the circumference of the base of the cone, which helps in calculating the radius of the base. 3. **Surface Area of Cone**: - This includes both the curved surface area and potentially the base (if stated in the problem). 4. **Volume of Cone**: - Utilize the volume formula for a cone, \( V = \frac{1}{3} \pi r^2 h \). Here, \( h \) can be derived using the Pythagorean theorem given the slant height and radius of the base. By following these steps systematically, we can determine both the surface area and volume of the cone formed from the given sector.
Transcribed Image Text:### Diagram Analysis and Problem The diagram represents a sector of a circular cardboard, with the following properties: - The length of the sides \( \overline{PQ} \) and \( \overline{PR} \) is 10 cm. - The angle \( \angle PQR \) is 126°. This sector is to be folded to form a cone such that \( \overline{PQ} \) and \( \overline{PR} \) coincide. #### Questions a) Calculate the surface area of the cone. b) Calculate the volume of the cone. - **Note**: Use \( \pi = \frac{22}{7} \). ### Graphical Explanation The diagram is a sector of a circle with: - Two straight lines \( \overline{PQ} \) and \( \overline{PR} \) of length 10 cm. - The included angle between these lines \( \angle PQR \) is 126°. When this sector is folded to form a cone: - The radius of the base of the cone can be derived from the arc length of the sector. - The slant height (10 cm) remains constant. ##### Computation: 1. **Arc Length Calculation**: - The arc length of the sector is part of the circumference of the circle from which it was cut. Using the formula for arc length, \( L = r \theta \) (where \( \theta \) is in radians), we calculate the length of the arc. 2. **Radius of Cone's Base**: - The arc length equals the circumference of the base of the cone, which helps in calculating the radius of the base. 3. **Surface Area of Cone**: - This includes both the curved surface area and potentially the base (if stated in the problem). 4. **Volume of Cone**: - Utilize the volume formula for a cone, \( V = \frac{1}{3} \pi r^2 h \). Here, \( h \) can be derived using the Pythagorean theorem given the slant height and radius of the base. By following these steps systematically, we can determine both the surface area and volume of the cone formed from the given sector.
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