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