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

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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.