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An illustration shows a 3 D triangular prism. The side of the base is 4.5 centimeters. The perpendicular height of the triangular base is 3.9 centimeters. The perpendicular height of the lateral face from the apex to the center of the base is 3.5 centimeters. There are two right angle symbols to show the perpendicular lines. The heights are shown as dashed lines.
A metalworker is creating decorative points in the form of triangular pyramids to go on top of a fence she is building. She wants to know the surface area of each pyramid. Use the surface area formula to find the surface area of each pyramid based on the figure.
(1 point)
___ cm2
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Base area = 1/2 · base · base-height = 1/2 · 4.5 · 3.9 = 8.775 cm².
For an equilateral base the inradius r = a·√3/6 = 4.5·√3/6 ≈ 1.299 cm. The slant height of a lateral face is
l = √(3.5² + r²) = √(3.5² + 1.299²) ≈ 3.734 cm.
Area of one lateral face = 1/2 · 4.5 · 3.734 ≈ 8.4015 cm², so three faces = 3 · 8.4015 ≈ 25.2045 cm².
Total surface area ≈ 8.775 + 25.2045 = 33.98 cm² ≈ 34.0 cm².
Answer: 34.0 cm² (≈34 cm²).
For an equilateral base the inradius r = a·√3/6 = 4.5·√3/6 ≈ 1.299 cm. The slant height of a lateral face is
l = √(3.5² + r²) = √(3.5² + 1.299²) ≈ 3.734 cm.
Area of one lateral face = 1/2 · 4.5 · 3.734 ≈ 8.4015 cm², so three faces = 3 · 8.4015 ≈ 25.2045 cm².
Total surface area ≈ 8.775 + 25.2045 = 33.98 cm² ≈ 34.0 cm².
Answer: 34.0 cm² (≈34 cm²).
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