A stairway runs up the edge of the
pyramid. From bottom to top the stairway
is 92 m long. The stairway makes an angle of 70° to the base edge, as shown. A line from the middle of one of the base edges to the top of the pyramid makes an angle of elevation of 52° with respect to the flat ground. Find the height of the pyramid.
I just can't figure out where 70 degrees comes into play, help is appreciated :)
The question in the textbook misses key information
Wait. I see it.
One face of the pyramid is an isosceles triangle with base angles of 70°
So, the altitude of one of the faces is 92 sin70°, and half the base of one face is 92 cos70°
so, if h is the pyramid height,
sin52° = h/(92sin70°)
h = 68.12
or,
tan52° = h/(92cos70°)
h = 40.27
Hmmm. I have messed up somewhere; maybe you can straighten it out.
Thanks so much!!
To find the height of the pyramid, we need to use the angles given in the problem and apply trigonometric functions. The angle of 70° is the angle the stairway makes with the base edge. This angle allows us to determine the length of the base edge of the pyramid.
Let's consider the right triangle formed by the base edge, the height of the pyramid, and the slant height of the pyramid (the length of the stairway).
Step 1: Finding the length of the base edge
Since we are given the length of the stairway (92 m) and the angle of 70°, we can use the trigonometric function cosine to find the length of the base edge.
Cosine relates the adjacent side to the hypotenuse of a right triangle. In this case, the adjacent side is the base edge and the hypotenuse is the stairway's length. Using the cosine function, we have:
cos(70°) = base edge / stairway's length
cos(70°) = base edge / 92
To find the base edge, we can rearrange the equation:
base edge = cos(70°) * 92
Now we know the length of the base edge.
Step 2: Finding the height of the pyramid
We have another right triangle formed by the height of the pyramid, the base edge (which we just found), and the line from the middle of one of the base edges to the top of the pyramid.
We are given that this line makes an angle of elevation of 52° with respect to the flat ground. This means that the angle between the ground and this line is 90° - 52° = 38°.
Using the tangent function, which relates the opposite side to the adjacent side in a right triangle, we can write:
tan(38°) = height of pyramid / base edge
Now, we can rearrange the equation to solve for the height of the pyramid:
height of pyramid = tan(38°) * base edge
Finally, we substitute the value we found for the base edge in the previous step to determine the height of the pyramid.
I can't see how this makes sense. Apparently the staircase runs up along one edge of the pyramid. Since the edge is longer than the altitude of one of the faces, its angle of elevation must be less.
Better describe the diagram. Be sure to mark and label all points and edges of interest.