Niko has an outdoor play tent in the form of a regular triangular pyramid, covered in fabric on all four sides. The surface area of the tent is 100, the base is 6, and the slant height is 8. What is the height of the base to the nearest tenth?
To find the height, we need to find the slant height of a lateral face. Let A be the area of the base triangle that is face ABC. Then we can use Heron's formula to find the area of this triangle: $\displaystyle A\ =\sqrt{s( s\ -\ a)( s\ -\ b)( s\ -\ c)}$, where $ s\ =\frac{a\ +\ b\ +\ c}{2}$, is the semiperimeter of the triangle with sides a, b, and c. But the base of the pyramid is 6 and the slant height is 8. Thus we have\[\displaystyle A\ =\sqrt{12( 12\ -\ 6)( 12\ -\ 8)( 12\ -\ 8)}\]
\[\displaystyle A\ =\sqrt{12( 6)( 4)( 4)}=\sqrt{(12)( 6\cdot 4)( 4)}\]
\[\displaystyle A\ =2\sqrt{(6)( 6\cdot 4)}=2\sqrt{(6)( 24)}\]
\[\displaystyle A\ =2\sqrt{144}=2( 12)=24\]The surface area of the tent is made up of 4 identical triangles, so\[\displaystyle 4\left(\frac{AB\cdot AC}{2}\right) =100\]Since the height of the triangle is the altitude to AB in triangle ABC, we can solve for the height using the area of a triangle formula. Using the area formula, we substitute 24 for the area of the triangle,\[\displaystyle \frac{6h}{2} =24\]to find $3h =24$, or h=8. Answer: \boxed{8}.