To solve the problem, let’s go through the steps in detail:
1. Find the radius of the original hemisphere:
- The diameter of the original hemisphere is 24 meters.
- The radius [tex]\( r \)[/tex] of a hemisphere is half of its diameter.
[tex]\[
r_{\text{original}} = \frac{24}{2} = 12 \text{ meters}
\][/tex]
2. Determine the scale factor and find the radius of the new hemisphere:
- The scale factor is given as [tex]\( k = 0.5 \)[/tex].
- The radius after dilation can be calculated by multiplying the original radius by the scale factor.
[tex]\[
r_{\text{new}} = 12 \times 0.5 = 6 \text{ meters}
\][/tex]
3. Calculate the surface area of a hemisphere:
- The surface area [tex]\( A \)[/tex] of a hemisphere is given by the formula:
[tex]\[
A = 2 \pi r^2
\][/tex]
4. Plug in the radius of the new hemisphere to find the surface area:
[tex]\[
A_{\text{new}} = 2 \pi (6)^2 = 2 \pi \times 36 = 72 \pi \text{ square meters}
\][/tex]
Therefore, the surface area of the new hemisphere is:
[tex]\[
\boxed{72 \pi \text{ m}^2}
\][/tex]
