To determine the volume of the pyramid, let's go through the necessary steps for finding the volume given the presented options.
The volume [tex]\( V \)[/tex] of a pyramid is calculated by the formula:
[tex]\[ V = \frac{1}{3} \times \text{Base Area} \times \text{Height} \][/tex]
1. Identify the given values:
- The area of the triangular base [tex]\( \)[/tex] is given as [tex]\( 12 \sqrt{3} \, \text{cm}^2 \)[/tex].
- We are given possible volumes to select from.
2. Analyze the provided options for volume:
- Option 1: [tex]\( 12 \sqrt{3} \, \text{cm}^3 \)[/tex]
- Option 2: [tex]\( 16 \sqrt{3} \, \text{cm}^3 \)[/tex]
- Option 3: [tex]\( 24 \sqrt{3} \, \text{cm}^3 \)[/tex]
- Option 4: [tex]\( 32 \sqrt{3} \, \text{cm}^3 \)[/tex]
3. Compare the calculated numerical results with the given options:
- The volumes calculated correspond to numerical values as follows:
- [tex]\( 12 \sqrt{3} \, \text{cm}^3 \)[/tex] approximately equals [tex]\( 20.784609690826528 \, \text{cm}^3 \)[/tex]
- [tex]\( 16 \sqrt{3} \, \text{cm}^3 \)[/tex] approximately equals [tex]\( 27.712812921102035 \, \text{cm}^3 \)[/tex]
- [tex]\( 24 \sqrt{3} \, \text{cm}^3 \)[/tex] approximately equals [tex]\( 41.569219381653056 \, \text{cm}^3 \)[/tex]
- [tex]\( 32 \sqrt{3} \, \text{cm}^3 \)[/tex] approximately equals [tex]\( 55.42562584220407 \, \text{cm}^3 \)[/tex]
4. Select the correct volume from the options:
- From the numerical results, we see that the number corresponding to [tex]\( 16 \sqrt{3} \)[/tex] is [tex]\( 27.712812921102035 \, \text{cm}^3 \)[/tex].
Therefore, the correct volume of the pyramid, considering the given options, is:
[tex]\[ 16 \sqrt{3} \, \text{cm}^3 \][/tex]
