A GREAT PYRAMID HAS A SQUARE BASE 230M ON EACH SIDE. THE 4 TRIANGULAR FACES EACH MAKE AN ANGLE OF 51.50' WITH THE GROUND.
1) HOW TALL IS THE PYRAMID ?
2) WHAT ARE THE DIMENSIONS OF EACH OF THESE TRIANGULAR FACES ?
the center of the pyramid is 115m from each edge. Find the height h using
h/115 = tan51°50'
Now figure the slant height s, using
s^2 = 115^2+h^2
The isosceles triangles thus have base 230 and altitude s.
The edges of the pyramid(the equal sides of the triangles) have length x, where
x^2 = s^2+115^2
To find the height of the pyramid, we can use trigonometry. The given information tells us that the triangular faces make an angle of 51.50' with the ground. This angle is between the base of the pyramid and one of the triangular faces.
1) To find the height, we can use the tangent function. The tangent of an angle is equal to the opposite side divided by the adjacent side. In this case, the height of the pyramid is the opposite side and the base of the pyramid is the adjacent side.
Let's calculate the height:
tan(angle) = opposite/adjacent
tan(51.50') = height/230m
Using a calculator, we find that tan(51.50') ≈ 1.2713
1.2713 = height/230m
Now, we can solve for the height:
height = 1.2713 * 230m
height ≈ 292.14m
Therefore, the height of the pyramid is approximately 292.14m.
2) The dimensions of each triangular face can be determined using the given information. Since the base of the pyramid is a square with sides measuring 230m, each triangular face will have the same dimensions.
The base of the triangular face will have a length equal to the side length of the square base, which is 230m.
The height of the triangular face will be the same as the height of the pyramid, which we calculated to be approximately 292.14m.
Therefore, the dimensions of each triangular face are 230m (base) and approximately 292.14m (height).
To find the answers to these questions, we can use trigonometry and the given information.
1) How tall is the pyramid?
To find the height of the pyramid, we can use the tangent function. The tangent of an angle in a right triangle is the ratio of the opposite side length to the adjacent side length.
In this case, the angle between the base of the pyramid and the ground is 51.50'. We can consider this angle as the angle between the base and one of the triangular faces. Let's call this angle α.
Let's first convert 51.50' to degrees:
51.50' = 51.50/60 degrees ≈ 0.8583 degrees
Now, let's find the height using the tangent function:
tan(α) = opposite side / adjacent side
tan(0.8583) = height / base
We know that the base of the pyramid is 230 meters on each side. Therefore, the equation becomes:
tan(0.8583) = height / 230
Now we can solve for the height:
height = tan(0.8583) * 230
Using a scientific calculator, we find that the height of the pyramid is approximately 66.9 meters.
Therefore, the pyramid is approximately 66.9 meters tall.
2) What are the dimensions of each of these triangular faces?
Since we know that the base of the pyramid is a square with each side measuring 230 meters, each triangular face has a base length equal to the side length of the square base. Therefore, the base length of each triangular face is 230 meters.
To find the other two sides of the triangular face, we can use the Pythagorean theorem, as the triangular face is a right triangle. One side is the height of the pyramid we found earlier, which is approximately 66.9 meters.
Using the Pythagorean theorem, we have:
(side length)^2 = (base length / 2)^2 + height^2
Substituting the values:
(side length)^2 = (230 / 2)^2 + (66.9)^2
Solving this equation will give us the side length of each triangular face.