A conical cup is made from a circular piece of paper with radius 10 cm by cutting out a sector and joining the edges as shown below. Suppose 0 = 97/5. r 10 cm 10 cm h 10 cm (a) Find the circumference C of the opening of the cup. C = 187 cm (b) Find the radius r of the opening of the cup. [Hint: Use C = 2Ar.] r =9 cm (c) Find the height h of the cup. [Hint: Use the Pythagorean Theorem.] h = 4.35 X cm (d) Find the volume V of the cup. V = 369.83 x cm3

Advanced Engineering Mathematics
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ISBN:9780470458365
Author:Erwin Kreyszig
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Chapter2: Second-order Linear Odes
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**Making a Conical Cup: Step-by-Step Guide**

A conical cup is constructed from a circular piece of paper with a radius of 10 cm by removing a sector and joining the edges. Suppose \(\theta = \frac{9\pi}{5}\).

### Illustrations:
1. **Circle and Sector**:
   - A circle with a radius labeled as 10 cm.
   - A sector is cut out, defined by an angle \(\theta\).

2. **Cone Formation**:
   - The resulting shape is a cone with:
     - Radius \( r \)
     - Height \( h \)
     - Slant height remaining as 10 cm (from the original circle).

### Calculations:

#### (a) Find the Circumference \( C \) of the Opening of the Cup.
- **Formula**: \( C = 18\pi \) cm

#### (b) Find the Radius \( r \) of the Opening of the Cup.
- **Given formula (hint)**: \( C = 2\pi r \)
- **Calculated Radius**: \( r = 9 \) cm

#### (c) Find the Height \( h \) of the Cup.
- **Hint**: Use the Pythagorean Theorem.
- **Calculated Height**: \( h = 4.35 \) cm (Note: this answer is incorrect based on provided checks).

#### (d) Find the Volume \( V \) of the Cup.
- Volume is calculated as \( V = 369.83 \) cm\(^3\)

For further details about each step and formulas used, you can visit our educational resources which elaborate on the mathematical concepts involved in creating geometric shapes from 2D surfaces.
Transcribed Image Text:**Making a Conical Cup: Step-by-Step Guide** A conical cup is constructed from a circular piece of paper with a radius of 10 cm by removing a sector and joining the edges. Suppose \(\theta = \frac{9\pi}{5}\). ### Illustrations: 1. **Circle and Sector**: - A circle with a radius labeled as 10 cm. - A sector is cut out, defined by an angle \(\theta\). 2. **Cone Formation**: - The resulting shape is a cone with: - Radius \( r \) - Height \( h \) - Slant height remaining as 10 cm (from the original circle). ### Calculations: #### (a) Find the Circumference \( C \) of the Opening of the Cup. - **Formula**: \( C = 18\pi \) cm #### (b) Find the Radius \( r \) of the Opening of the Cup. - **Given formula (hint)**: \( C = 2\pi r \) - **Calculated Radius**: \( r = 9 \) cm #### (c) Find the Height \( h \) of the Cup. - **Hint**: Use the Pythagorean Theorem. - **Calculated Height**: \( h = 4.35 \) cm (Note: this answer is incorrect based on provided checks). #### (d) Find the Volume \( V \) of the Cup. - Volume is calculated as \( V = 369.83 \) cm\(^3\) For further details about each step and formulas used, you can visit our educational resources which elaborate on the mathematical concepts involved in creating geometric shapes from 2D surfaces.
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