Sure, let's break down the solution step-by-step:
1. Identify the Given Values:
- Altitude of the pyramid (height from the apex to the base) is 9 cm.
- Side length of the base hexagon is 3 cm.
- Distance from the base to the horizontal cross-section (height at which the cross-section is taken) is 5 cm.
2. Determine the Height from the Apex to the Cross-Section:
- Since the cross-section is 5 cm above the base, the height from the apex to the cross-section is [tex]\(9 \text{ cm} - 5 \text{ cm} = 4 \text{ cm}\)[/tex].
3. Calculate the Scale Factor:
- The scale factor for similar triangles is the ratio of the height from the apex to the cross-section to the total height of the pyramid.
- Hence, the scale factor is:
[tex]\[
\text{Scale Factor} = \frac{\text{Height from apex to cross-section}}{\text{Total height}} = \frac{4}{9} \approx 0.4444
\][/tex]
4. Determine the Side Length of the Hexagon in the Cross-Section:
- The side length of the hexagon in the cross-section is scaled by the same factor as the height.
- Therefore, the side length of the cross-section hexagon is:
[tex]\[
\text{Cross-Section Side} = \text{Base Side} \times \text{Scale Factor} = 3 \text{ cm} \times 0.4444 \approx 1.3333 \text{ cm}
\][/tex]
5. Calculate the Area of the Hexagon:
- The formula to calculate the area of a regular hexagon with side length [tex]\( a \)[/tex] is:
[tex]\[
\text{Area} = \frac{3 \sqrt{3}}{2} \times a^2
\][/tex]
- Substituting [tex]\( a = 1.3333 \text{ cm} \)[/tex] into this formula, we get:
[tex]\[
\text{Area} \approx \frac{3 \sqrt{3}}{2} \times (1.3333)^2 \approx 4.6188 \text{ cm}^2
\][/tex]
Thus, the area of the horizontal cross-section 5 cm above the base is approximately [tex]\( 4.6188 \text{ cm}^2 \)[/tex].
