If you unfurl the lateral "face" of a cone, you get the sector of a circle whose central angle subtends an arc with length [tex]L[/tex] equal to the circumference of the cone's base, and a radius equal to the slant height [tex]s[/tex] of the cone.
Since the diameter of the base is 18m, the circumference of the base is about [tex]3.14\times18=56.52[/tex] meters, so this is the value of [tex]L[/tex].
You're given a slant height of [tex]s=15[/tex] meters.
Now, the following proportional relation holds for any circle:
[tex]\dfrac{\text{area of sector}}{\text{area of circle}}=\dfrac{\text{arc length of sector}}{\text{circumference of circle}}[/tex]
This translates to
[tex]\dfrac A{\pi\times s9^2}=\dfrac s{2\pi s\times9}[/tex]
(remember, we're talking about a sector of a circle with radius [tex]s[/tex] and arc length [tex]18\pi[/tex]). Solving for [tex]A[/tex]:
[tex]\dfrac A{81\times3.14}=\dfrac{15}{18\times15\times3.14}[/tex]
[tex]A=\dfrac92=4.5[/tex]
So the area of the lateral face is approximately 4.5 square meters.
