Question

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.

Answer 1

looking perpendicular to one face of the pyramid (base a square of side b, and height h),

h/(b/2) = tan 52°

looking up the edge,

h/√((b^2/2 + h^2) = sin 70°

Clean things up a bit, and you can solve for b and h

Answer 2

It occurs to me that I may have misinterpreted the unseen diagram. If the 70° angle is on the face of the pyramid, then we have

h/(b/2) = tan 52°

and the slant height s is in the other triangle, rather than just h, so

√((b/2)^2+h^2)/√(b^2/2 + h^2) = sin 70°

Answer 3

You can solve this by using trigonometry.
One face of the pyramid is an isosceles triangle with base angles of 70°

First, you try to find the altitude of the isosceles triangle which is the line from the middle of one of the base edges to the top of the pyramid.
Sin 70° = x/92
x = 86.5m

Now you can solve for the height, h
The altitude of the isosceles triangle is the hypotenuse and the angle of elevation is 52°
using sin we can solve for the height.
Sin 52° = x/86.5
x = 68.1m

Answer 4

Sin 70° = x/92
x = 86.5m

Sin 52° = x/86.5
x = 68.1m

Answer 5

No