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A hemisphere with a diameter of 24 meters is dilated with a scale factor of [tex]k=0.5[/tex]. What is the surface area of the new hemisphere?

A. [tex]72 \pi \, \text{m}^2[/tex]
B. [tex]144 \pi \, \text{m}^2[/tex]
C. [tex]288 \pi \, \text{m}^2[/tex]
D. [tex]576 \pi \, \text{m}^2[/tex]



Answer :

GinnyAnswer
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]

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