Question

The base of a pyramid ABCDE is rectangle ABCD with width CD = 15 cm. The apex of the pyramid, E, is directly above the center of the base point O. The angle between lateral face AABE and the base is 54° while the angle between lateral face ABCE and the base is 73°. Find the volume of the pyramid. Round your answer to the cm³ m P nearest m M​

Answer 1

4372 cm³

Explanation:

You want the volume of a symmetrical right rectangular pyramid whose lateral faces make angles of 73° and 54° with the base, and whose short side is 15 cm.

Height

The height of the pyramid can be found using the tangent relation. The distance from the center to the steep face is (15 cm)/2 = 7.5 cm. The height will satisfy ...

Tan = Opposite/Adjacent

tan(73°) = height/(7.5 cm)

height = (7.5 cm)tan(73°) ≈ 24.5314 cm

Side

A similar relation can be used to find the longer side length:

tan(54°) = height/(side/2)

side = 2·height/tan(54°) = (15 cm)tan(73°)/tan(54°) ≈ 35.6462 cm

Volume

The volume is ...

V = 1/3Bh

V = 1/3(15 cm)(35.6462 cm)(24.5314 cm) ≈ 4372 cm³

The volume of the pyramid is about 4372 cm³.

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