To find the height of the triangular base of Niko's outdoor play tent (a regular triangular pyramid), we can use the information about the surface area, base, and slant height.
The surface area (SA) of a triangular pyramid is given by the formula:
\[ SA = \text{Base Area} + \text{Lateral Area} \]
The base of the tent is a triangle with a base length of \( b = 6 \) ft. To find the area of the triangular base, we need its height \( h_b \):
\[ \text{Base Area} = \frac{1}{2} \times b \times h_b = \frac{1}{2} \times 6 \times h_b = 3h_b \]
For the lateral area, since the pyramid is regular, there are three triangular lateral faces. Each face has a base of 6 ft and a height equal to the slant height (8 ft):
\[ \text{Lateral Area} = \frac{1}{2} \times \text{base} \times \text{slant height} \times 3 = \frac{1}{2} \times 6 \times 8 \times 3 = 72 \text{ ft}^2 \]
Now we can write the equation for the total surface area:
\[ SA = \text{Base Area} + \text{Lateral Area} \] \[ 100 = 3h_b + 72 \]
Now, solve for \( h_b \):
\[ 100 - 72 = 3h_b \] \[ 28 = 3h_b \] \[ h_b = \frac{28}{3} \approx 9.3 \text{ ft} \]
Thus, the height of the base of the tent, to the nearest tenth, is \( 9.3 \text{ ft} \).
So the correct answer is: 9.3 ft.
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