Niko has an outdoor play tent in the form of a regular triangular pyramid, covered fabric on all four sides. The surface area is 100 ft.^2, the base is 6 ft., and the slant height is 8 ft. What is the height of the base to the nearest tenth.
First, we find the area of one of the triangular sides. This is $100/4=25$ square feet. If the side length of one of the triangles were $x$, we could find the area by Heron's formula. Heron's formula tells us that the area of a triangle with side lengths $a$, $b$, and $c$ is $\sqrt{s(s-a)(s-b)(s-c)}$, where $s=\frac{a+b+c}{2}$. In this case, $a=x$, $b=x$, and $c=6$. We have that $s=(x+x+6)/2=(2x+6)/2=x+3$, so we have \[25=\sqrt{(x+3)(x+3-x)(x+3-x)(x+3-6)}=\sqrt{(x+3)(3-x)(3-x)(-3+x)} =\sqrt{(9-x^2)(9-x^2)} =9-x^2.\] We have the equation $25=9-x^2$, so $34=x^2$, and $x=\sqrt{34}$. The height is the solution to the problem, or $\boxed{5.8\text{ feet}}$.
Niko has an outdoor play tent in the form of a regular triangular pyramid, covered fabric on all four sides. The surface area is 100 ft.^2, the base is 6 ft., and the slant height is 8 ft. What is the height of the base to the nearest tenth.
A. 9.3 ft
B. 6.5 ft
C. 57.3 ft
D. 372 ft
The surface area of the pyramid is equal to the sum of the areas of its triangular faces. Since the tent has four sides, each triangular face has an area of 25 ft².
Since the base of the triangle is 6 ft, the height of the triangular face can be found using the formula for the area of a triangle: $\frac{1}{2}bh$.
Thus, we have $\frac{1}{2}(6)(h) = 25$, which simplifies to $3h = 25$ and $h = \frac{25}{3}$ ft.
We can use the Pythagorean theorem to find the slant height $l$ of the triangular face, using the height $h$ and the base $6$ ft.
$l^2 = h^2 + (\frac{6}{2})^2 = h^2 + 9$
$l^2 = (\frac{25}{3})^2 + 9 = \frac{625}{9} + 9 = \frac{625 + 81}{9} = \frac{706}{9}$
$l \approx \sqrt{\frac{706}{9}} \approx \sqrt{\frac{700}{9}} \approx \frac{\sqrt{700}}{\sqrt{9}} = \frac{10\sqrt{7}}{3}$ ft.
Finally, we can use the Pythagorean theorem to find the height $H$ of the pyramid, using the slant height $l$ and the height of the triangular face $h$.
$H^2 = l^2 - h^2 = (\frac{10\sqrt{7}}{3})^2 - (\frac{25}{3})^2 = \frac{100(7)}{9} - \frac{625}{9} = \frac{700 - 625}{9}$
$H = \sqrt{\frac{700 - 625}{9}} = \sqrt{\frac{75}{9}} = \sqrt{\frac{25}{3}} = \frac{\sqrt{25}}{\sqrt{3}} = \frac{5}{\sqrt{3}} = \frac{5\sqrt{3}}{3}$ ft.
To the nearest tenth, the height of the base is $\approx \boxed{5.8}$ ft.