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.
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
Sin 70° = x/92
x = 86.5m
Sin 52° = x/86.5
x = 68.1m
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
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°
Why did the pyramid go to school?
To get a higher education!
Now, let's solve your problem. To find the height of the pyramid, we can use the angle of elevation and some trigonometry. The line from the middle of one of the base edges to the top of the pyramid forms a right triangle with the ground. The angle of elevation is the angle between the ground and the line connecting the middle of the base to the top.
Given that the angle of elevation is 52°, we can use the tangent function.
Let's call the height of the pyramid 'h'. The opposite side of the triangle is 'h' and the adjacent side is half of the base length (92 m/2 = 46 m).
Using the tangent function:
tan(52°) = h/46 m
Now, we can solve for 'h':
h = 46 m * tan(52°)
And there you have it! Plug in those numbers into a calculator and you'll find the height of the pyramid.
To find the height of the pyramid, we can use trigonometry.
Let's consider a right triangle formed by the height of the pyramid, the distance from the middle of one of the base edges to the top of the pyramid, and the distance from the middle of one of the base edges to the bottom of the stairway.
First, let's find the length of the base edge of the pyramid. Since the stairway is 92 m long and makes an angle of 70° with the base edge, we can use the cosine function to calculate it.
cos(70°) = (length of base edge) / (length of stairway)
Rearranging the formula, we get:
(length of base edge) = (length of stairway) / cos(70°)
Substituting the given values, we have:
(length of base edge) = 92 m / cos(70°)
Using a calculator to evaluate cos(70°), we find that cos(70°) is approximately 0.3420.
(length of base edge) = 92 m / 0.3420
(length of base edge) ≈ 269.59 m
Now, let's find the length of the side adjacent to the angle of elevation. This side is the distance from the middle of one of the base edges to the top of the pyramid.
Using trigonometry, we can use the cosine function again:
cos(52°) = (distance to the top) / (length of base edge)
Rearranging the formula, we have:
(distance to the top) = (length of base edge) * cos(52°)
Substituting the previously calculated length of the base edge, we get:
(distance to the top) = 269.59 m * cos(52°)
Using a calculator to evaluate cos(52°), we find that cos(52°) is approximately 0.6157.
(distance to the top) = 269.59 m * 0.6157
(distance to the top) ≈ 165.99 m
Finally, we have the height of the pyramid, which is the side opposite to the angle of elevation. Using the Pythagorean theorem, we can find it:
(height of the pyramid)^2 = (length of the base edge)^2 - (distance to the top)^2
Substituting the values we found earlier, we have:
(height of the pyramid)^2 = (269.59 m)^2 - (165.99 m)^2
Simplifying the equation, we get:
(height of the pyramid)^2 ≈ 72682.48 m^2 - 27590.20 m^2
(height of the pyramid)^2 ≈ 45092.28 m^2
Taking the square root of both sides, we find:
height of the pyramid ≈ sqrt(45092.28 m^2)
height of the pyramid ≈ 212.48 m
Therefore, the height of the pyramid is approximately 212.48 meters.