To find the height of the triangular base (h), we first need to establish some geometric relationships for a regular triangular pyramid (tetrahedron).
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The surface area (SA) of a triangular pyramid can be calculated using the formula:
\[ SA = \text{Base Area} + \text{Lateral Area} \]
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The base is a triangle, and its area can be calculated using the base length (b) and height (h) of the triangle:
\[ \text{Area of Base} = \frac{1}{2} \times b \times h \]
In this case, the base length \( b = 6 \) ft.
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The lateral area consists of the areas of the three triangular faces. The area of one triangular face can be calculated using the slant height (l) and the base edge (b):
\[ \text{Area of One Lateral Face} = \frac{1}{2} \times b \times l \]
Therefore, the total lateral area (for three faces) is:
\[ \text{Total Lateral Area} = 3 \times \frac{1}{2} \times b \times l = \frac{3}{2} \times b \times l \]
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We know the slant height \( l = 8 \) ft, and we can now calculate the areas:
- Base area: \[ \text{Area of Base} = \frac{1}{2} \times 6 \times h = 3h \]
- Total lateral area: \[ \text{Total Lateral Area} = \frac{3}{2} \times 6 \times 8 = 72 \text{ ft}^2 \]
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Therefore, we can set up the equation for the total surface area: \[ SA = \text{Area of Base} + \text{Total Lateral Area} \] \[ 100 = 3h + 72 \]
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Now, solve for \( h \): \[ 100 - 72 = 3h \] \[ 28 = 3h \] \[ h = \frac{28}{3} \approx 9.33 \text{ ft} \]
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Rounding \( h \) to the nearest tenth, we get: \[ h \approx 9.3 \text{ ft} \]
So, the height of the base is 9.3 ft.