What angles do lateral faces make with the base plane of a rectangular pyramid with base edges of 10 m and 16 m and with lateral edges of 18 m each? Please help!
The vertex of the pyramid lies above a point which is 5m from the long sides of the base, and 8m from the short sides.
So, we can compute the height of the pyramid using
h^2 + 5^2+8^2 = 18^2
Now, using h, we can find the angles using either
tanθ = h/8 or tanθ = h/5
To find the angles between the lateral faces and the base plane of a rectangular pyramid, we can apply trigonometry concepts.
Step 1: Visualize the Pyramid
A rectangular pyramid has a rectangular base and four triangular lateral faces that meet at the apex. In this case, the base has edges of 10 m and 16 m, and the lateral edges (slant height) are 18 m each.
Step 2: Find the Height
To find the height of the pyramid, we can use the Pythagorean theorem. Since the base is a rectangle, the height is one of the sides of the right triangle formed by the diagonal of the rectangle and one of the base edges. Let's call this height h.
Using the Pythagorean theorem:
h^2 + (10/2)^2 = 16^2
h^2 + 5^2 = 16^2
h^2 + 25 = 256
h^2 = 256 - 25
h^2 = 231
h = sqrt(231)
h ≈ 15.2 m
Step 3: Calculate the Angles
Now that we have the height, we can calculate the angles between the lateral faces and the base plane.
Consider one of the triangular lateral faces. It is a right triangle with one side being the height (h = 15.2 m), another side being the lateral edge (18 m), and the hypotenuse being the slant height of the triangle.
Using trigonometry:
cos(angle) = adjacent / hypotenuse
cos(angle) = h / lateral edge
cos(angle) ≈ 15.2 / 18
angle ≈ arccos(15.2 / 18)
Repeat this calculation for each of the four lateral faces to find all the angles they make with the base plane.
Note: Make sure your calculator is in degree mode when finding the inverse cosine (arccos) function.
I hope this explanation helps!