Question
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 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?
To find the height of the triangular pyramid, we need to find the height of the triangle and then apply the Pythagorean theorem.
The surface area of a triangular pyramid is given by the formula:
Surface Area = base area + lateral area.
Since the tent is covered in fabric on all four sides, the surface area is equal to the lateral area.
The lateral area of a triangular pyramid can be found using the formula:
Lateral Area = (1/2) * p * s,
where p is the perimeter of the base and s is the slant height.
The slant height is given as 8 ft. To find the perimeter of the base, we need to know the length of one side of the base of the triangular pyramid.
However, it is not clear whether the given base length of 6 ft refers to the length of one side of the base or the sum of the lengths of all three sides of the base.
If the base length given is the sum of the lengths of all three sides (perimeter), we can find the length of one side by dividing the base length by 3:
One side of the base = 6 ft / 3 = 2 ft.
Plugging these values into the formula for lateral area, we get:
Lateral Area = (1/2) * (2 ft + 2 ft + 2 ft) * 8 ft = 12 ft * 8 ft = 96 ft².
However, if the base length given is the length of one side of the base, we already have all the required values to find the lateral area:
Lateral Area = (1/2) * (6 ft + 6 ft + 6 ft) * 8 ft = 18 ft * 8 ft = 144 ft².
Since the surface area of the tent is given as 100 ft², we can conclude that the base length given is the length of one side of the base (perimeter of the base is not equal to 6 ft).
Now, to find the height, we can use the Pythagorean theorem, which states that in a right triangle, the square of the length of the hypotenuse (slant height) is equal to the sum of the squares of the lengths of the other two sides.
In this case, the height of the pyramid is one side, the slant height is the hypotenuse, and the base is the other side.
Applying the Pythagorean theorem:
height² + (base/2)² = slant height²,
height² + (6 ft/2)² = 8 ft²,
height² + 3 ft² = 64 ft²,
height² = 64 ft² - 3 ft²,
height² = 61 ft².
To find the height, we take the square root of both sides:
height ≈ √61 ft ≈ 7.8 ft.
Therefore, the height of the triangular pyramid (and the base) is approximately 7.8 ft to the nearest tenth.