Let's solve the problem step-by-step:
1. Given Data and Information:
- Radius of the sector (which becomes the slant height of the cone): [tex]\( r = 14 \)[/tex] cm
- Angle of the sector: [tex]\( \theta = 90^\circ \)[/tex]
2. Determine the Arc Length of the Sector:
The arc length (L) of the sector can be calculated using the formula:
[tex]\[
L = 2 \pi r \left( \frac{\theta}{360} \right)
\][/tex]
Plugging in the values:
[tex]\[
L = 2 \pi \cdot 14 \left( \frac{90}{360} \right) = 2 \pi \cdot 14 \left( \frac{1}{4} \right) = \pi \cdot 7
\][/tex]
Numerically:
[tex]\[
L \approx 21.991148575128552 \text{ cm}
\][/tex]
3. Find the Radius of the Cone's Base:
The radius [tex]\(R\)[/tex] of the base of the cone is the same as the arc length divided by [tex]\(2\pi\)[/tex]:
[tex]\[
R = \frac{L}{2\pi} = \frac{21.991148575128552}{2\pi}
\][/tex]
Numerically:
[tex]\[
R \approx 3.5 \text{ cm}
\][/tex]
4. Slant Height of the Cone:
The slant height [tex]\(s\)[/tex] of the cone is given by the radius of the original sector:
[tex]\[
s = 14 \text{ cm}
\][/tex]
5. Lateral (Curved) Surface Area of the Cone:
The lateral surface area [tex]\(A\)[/tex] of the cone can be determined using the formula:
[tex]\[
A = \pi R s
\][/tex]
Plugging in the values:
[tex]\[
A = \pi \cdot 3.5 \cdot 14 \approx 153.93804002589985 \text{ cm}^2
\][/tex]
Therefore, the lateral surface area of the cone is approximately [tex]\( 153.938 \text{ cm}^2 \)[/tex].
