Jacob is going camping and is setting up a tent in the form of a regular triangular pyramid, covered in fabric on all four sides. The surface area of the tent is 65 ft2, the base is 4 ft, and the slant height is 7 ft. What is the height of the base to the nearest tenth.
A. 1.5 ft
B. 115 ft
C. 21 ft
D. 11.5 ft
We know that the surface area of the triangular pyramid is given by the formula $SA=\frac{1}{2}pl+bs$, where $p$ is the perimeter of the base, $l$ is the slant height, $b$ is the base, and $s$ is the height of each of the triangular faces. Since there are four triangular faces, the surface area is given by $65=2(4)(7)+4s$. Solving for $s$ gives $s=7-4=3$. Using the Pythagorean Theorem on one of the triangular faces, we have $\sqrt{3^2+h^2}=7$. Solving gives $h^2=7^2-3^2=40$, so $h=\sqrt{40}=2\sqrt{10}\approx\boxed{6.3\text{ ft}}$.
bot, that is confusing
Apologies for the confusion. Let's approach the problem differently:
A regular triangular pyramid has an equilateral triangle as its base. In this case, the base has a length of 4 ft. The slant height, which is the distance from the top of the pyramid to the edge of the base triangle, is given as 7 ft.
To find the height of the pyramid, which is the distance from the top of the pyramid to the center of the base, we can use the Pythagorean theorem. Since the base triangle is equilateral, we can draw a perpendicular line from the center of the base to one of the sides, splitting the base triangle into two congruent right triangles.
Let's call the height of the pyramid 'h'. The length of one side of the base triangle is 4 ft, so the length of the line segment from the center of the base to one of the edges is half of that, which is 2 ft. We can label this segment as 'a'. The slant height of the pyramid is 7 ft, so again utilizing the equilateral nature of the base triangle, we can label the other side of the right triangle as 'b'.
Using the Pythagorean theorem, we have:
$a^2 + b^2 = h^2$
$2^2 + 7^2 = h^2$
$4 + 49 = h^2$
$53 = h^2$
Taking the square root of both sides, we get:
$h = \sqrt{53} \approx 7.3$ ft
Therefore, the height of the pyramid, from the top to the center of the base, is approximately 7.3 ft.