NOT TO SCALE $\newline$ $A O B$ is a sector of a circle, centre $O$ . $\newline$ $A O=10 \mathrm{~cm}$ and the sector angle is $216^{\circ}$ . $\newline$ (i) Calculate the length of the are of this sector. $\newline$ Give your answer as a multiple of $\pi$ . $\newline$ (ii) A cone is made from this sector by joining $O A$ to $O B$ . $\newline$ Calculate the volume of the cone. $\newline$ [The volume, $V$ , of a cone with radius $r$ and height $h$ is $O$ $0$ .]

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Q. NOT TO SCALE

\newline

A O B

is a sector of a circle, centre

O

\newline

A O=10 \mathrm{~cm}

and the sector angle is

216^{\circ}

\newline

(i) Calculate the length of the are of this sector.

\newline

Give your answer as a multiple of

\pi

\newline

(ii) A cone is made from this sector by joining

O A

O B

\newline

Calculate the volume of the cone.

\newline

[The volume,

V

, of a cone with radius

r

and height

h

O

0

Use Arc Length Formula: To find the length of the arc, use the formula for the arc length of a circle, which is $(\theta/360) \times 2\pi r$ , where $\theta$ is the angle in degrees and $r$ is the radius.
Plug in Given Values: Plug in the given values: $\theta = 216$ degrees and $r = 10$ cm. $\newline$ Arc length = $(216/360) \times 2\pi \times 10$ .
Simplify Fraction: Simplify the fraction $\frac{216}{360}$ to $\frac{3}{5}$ . $\newline$ Arc length = $\left(\frac{3}{5}\right) \times 2\pi \times 10$ .
Multiply Numbers: Multiply the numbers together. $\newline$ Arc length = $6\pi \times 10$ .
Calculate Arc Length: Calculate the arc length. Arc length = $60\pi$ cm.
Find Cone Volume: Now, calculate the volume of the cone. The radius of the cone is the same as the radius of the sector, which is $10\,\text{cm}$ . The slant height of the cone is also the radius of the sector, but we need the perpendicular height for the volume formula.
Convert Angle to Radians: To find the height of the cone, we can use the formula for the area of a sector, which is $(1/2) \times r^2 \times \theta$ in radians. But we need to convert the angle from degrees to radians first.
Use Area of Sector Formula: Convert the angle to radians by multiplying by $\pi/180$ . $\newline$ $\theta$ in radians = $216 \times (\pi/180)$ .
Calculate Area: Simplify the expression. $\theta$ in radians = $1.2\pi$ .
Use Lateral Surface Area Formula: Now, use the area of the sector formula: Area = $(1/2) \times r^2 \times \theta$ . Area = $(1/2) \times 10^2 \times 1.2\pi$ .
Solve for Radius: Calculate the area. $\newline$ Area = $(\frac{1}{2}) \times 100 \times 1.2\pi$ .
Simplify Expression: Area = $60\pi \, \text{cm}^2$ .
Simplify Expression: Area = $60\pi \,\text{cm}^2$ .The area of the sector is equal to the lateral surface area of the cone. To find the height of the cone, we need to use the formula for the lateral surface area of a cone, which is $\pi rl$ , where $l$ is the slant height.
Simplify Expression: Area = $60\pi \,\text{cm}^2$ .The area of the sector is equal to the lateral surface area of the cone. To find the height of the cone, we need to use the formula for the lateral surface area of a cone, which is $\pi rl$ , where $l$ is the slant height.We know the lateral surface area ( $60\pi$ ) and the slant height ( $10 \,\text{cm}$ ), so we can solve for the radius $r$ of the cone. $\newline$ $60\pi = \pi r \times 10$ .
Simplify Expression: Area = $60\pi \, \text{cm}^2$ . The area of the sector is equal to the lateral surface area of the cone. To find the height of the cone, we need to use the formula for the lateral surface area of a cone, which is $\pi rl$ , where $l$ is the slant height. We know the lateral surface area ( $60\pi$ ) and the slant height ( $10 \, \text{cm}$ ), so we can solve for the radius $r$ of the cone. $\newline$ $60\pi = \pi r \times 10$ . Solve for $r$ . $\newline$ $r = \frac{60\pi}{\pi \times 10}$ .
Simplify Expression: Area = $60\pi \, \text{cm}^2$ . The area of the sector is equal to the lateral surface area of the cone. To find the height of the cone, we need to use the formula for the lateral surface area of a cone, which is $\pi rl$ , where $l$ is the slant height. We know the lateral surface area ( $60\pi$ ) and the slant height ( $10 \, \text{cm}$ ), so we can solve for the radius $r$ of the cone. $\newline$ $60\pi = \pi r \times 10$ . Solve for $r$ . $\newline$ $r = \frac{60\pi}{\pi \times 10}$ . Simplify the expression. $\newline$ $r = 6 \, \text{cm}$ .