a cone is made by Rolling a metal sheet in the form of a sector with radius 36 CM are the central angle 150 degree find the curved surface area and the volume of a cone so formed

To find the **curved surface area** and **volume** of the cone formed from a sector, we need to work through the problem step by step.

### Given:
- Radius of the sector (the slant height of the cone, \( l \)) = 36 cm
- Central angle of the sector = 150°

#### Step 1: Finding the radius of the cone (\(r\))
The length of the arc of the sector becomes the circumference of the base of the cone.

- **Arc length** of the sector = \( \frac{\theta}{360^\circ} \times 2\pi l \)

Where:
- \( \theta \) is the central angle of the sector (150°)
- \( l \) is the slant height (36 cm)

Substitute the values:

\[
\text{Arc length} = \frac{150^\circ}{360^\circ} \times 2\pi \times 36 = \frac{5}{12} \times 2\pi \times 36 = 30\pi \, \text{cm}
\]

Now, the arc length becomes the **circumference** of the cone's base:

\[
\text{Circumference of base of the cone} = 2\pi r
\]

Equating the arc length and the circumference:

\[
30\pi = 2\pi r
\]

Simplifying:

\[
r = \frac{30\pi}{2\pi} = 15 \, \text{cm}
\]

Thus, the **radius of the cone's base** is \( r = 15 \, \text{cm} \).

#### Step 2: Finding the height of the cone (\(h\))
The slant height \( l \), radius \( r \), and height \( h \) of the cone are related by the Pythagorean theorem:

\[
l^2 = r^2 + h^2
\]

Substitute the known values:

\[
36^2 = 15^2 + h^2
\]

\[
1296 = 225 + h^2
\]

\[
h^2 = 1296 - 225 = 1071
\]

\[
h = \sqrt{1071} \approx 32.72 \, \text{cm}
\]

So, the height of the cone is approximately \( h = 32.72 \, \text{cm} \).

#### Step 3: Finding the **curved surface area** of the cone
The curved surface area \( A \) of the cone is given by the formula:

\[
A = \pi r l
\]

Substitute the values of \( r = 15 \, \text{cm} \) and \( l = 36 \, \text{cm} \):

\[
A = \pi \times 15 \times 36 = 540\pi \, \text{cm}^2
\]

So, the **curved surface area** of the cone is \( 540\pi \, \text{cm}^2 \), or approximately \( 1696.46 \, \text{cm}^2 \).

#### Step 4: Finding the **volume** of the cone
The volume \( V \) of the cone is given by the formula:

\[
V = \frac{1}{3} \pi r^2 h
\]

Substitute the values of \( r = 15 \, \text{cm} \) and \( h = 32.72 \, \text{cm} \):

\[
V = \frac{1}{3} \pi \times 15^2 \times 32.72 = \frac{1}{3} \pi \times 225 \times 32.72
\]

\[
V = \frac{1}{3} \times 225 \times 32.72 \times \pi = 2454\pi \, \text{cm}^3
\]

Thus, the **volume** of the cone is approximately \( 7707.89 \, \text{cm}^3 \).

### Final Answer:
- **Curved surface area** = \( 540\pi \, \text{cm}^2 \) ≈ 1696.46 cm²
- **Volume** = \( 2454\pi \, \text{cm}^3 \) ≈ 7707.89 cm³

---

Would you like more details on any step?

Here are some related questions:
1. What is the formula for the total surface area of a cone?
2. How does the sector's angle affect the dimensions of the cone?
3. Can a sector with a different angle form a different cone?
4. What are the properties of a cone formed by a right circular sector?
5. How can we find the slant height if the radius and height are known?

**Tip:** For cones, the slant height is always the longest side when looking at the cross-section triangle formed by height and radius.

A cone is made by rolling a metal sheet in the form of a sector with radius 36 cm and a central angle of 150 degrees. Find the curved surface area and the volume of the cone so formed.

Learn how to calculate the curved surface area and volume of a cone formed by rolling a sector with a radius of 36 cm and a 150-degree central angle. This problem involves geometry concepts like arc length, curved surface area, and volume calculation, using formulas such as πrl and (1/3)πr^2h. Suitable for Grades 9-11 students.

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