To find the height of the triangular pyramid (also known as a tetrahedron or triangular prism with a square base), we can start with the given information.
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Surface Area of the Pyramid: The total surface area \( S \) of the pyramid consists of the area of the base and the area of the three triangular faces: \[ S = \text{Area of Base} + \text{Area of 3 Triangular Faces} \] Given that the surface area \( S = 100 , \text{ft}^2 \) and the base is a triangle with side length \( b = 6 , \text{ft} \).
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Area of the Base (Triangle): The area \( A \) of an equilateral triangle is given by: \[ A = \frac{\sqrt{3}}{4} b^2 \] Substituting for \( b = 6 \): \[ A = \frac{\sqrt{3}}{4} \times 6^2 = \frac{\sqrt{3}}{4} \times 36 = 9\sqrt{3} \approx 15.588 , \text{ft}^2 \]
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Area of the 3 Triangular Faces: The combined area of the three triangular faces is: \[ \text{Area of 3 Triangular Faces} = S - \text{Area of Base} = 100 - 15.588 \approx 84.412 , \text{ft}^2 \]
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Area of One Triangular Face: Each triangular face of the pyramid is a triangle with base 6 ft and slant height \( l = 8 , \text{ft} \). The area of one triangular face can be found using: \[ \text{Area}{\text{triangle}} = \frac{1}{2} \times \text{base} \times \text{height} \] The height of the triangular face in this case is the slant height, which gives: \[ \text{Area}{\text{one triangular face}} = \frac{1}{2} \times 6 \times 8 = 24 , \text{ft}^2 \]
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Total Area of the Three Triangular Faces: \[ \text{Area of 3 Faces} = 3 \times \text{Area}_{\text{one triangular face}} = 3 \times 24 = 72 , \text{ft}^2 \]
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Finding the Height of the Base: Since we are given the total surface area, we arrive at a discrepancy because we calculated the area of the triangular faces as \( 72 , \text{ft}^2 \). This indicates there are errors in assumptions or understanding of the geometry.
To resolve the height, we apply the slant height formula.
In a triangular pyramid, the slant height, the height of the triangle, and the half base forms a right triangle: Using Pythagorean theorem, where the slant height is 8 ft and half base is: \[ \frac{6}{2} = 3 , \text{ft}, \] So, we use: \[ h^2 + 3^2 = 8^2, \] \[ h^2 + 9 = 64, \] \[ h^2 = 64 - 9 = 55, \] \[ h = \sqrt{55} \approx 7.4 , \text{ft}. \]
Thus, the height of the base to the nearest tenth is approximately: \[ \text{Height} \approx 7.4 , \text{ft}. \] None of the given responses matches 7.4 ft exactly. Still, the closest approximation assuming some slight tolerance suggests that the provided estimate might best use rounding toward allowable approximations. The most realistic pick in context should have been an adaptive response based on plotted inputs.
So, if tallied correctly, the ultimate output should come in line with provided estimates graphed under part calculations yet require review for truncation matching applied with initial inputs.
Conclusion:
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