Asked by julie
A solid oblique pyramid has an equilateral triangle as a base with an edge length of 4StartRoot 3 EndRoot cm and an area of 12StartRoot 3 EndRoot cm2.
A solid oblique pyramid has an equilateral triangle base with an edge length of 4 StartRoot 3 EndRoot centimeters and an area of 12 StartRoot 3 EndRoot centimeters squared. The triangle base has points A C D and the apex is point B. The angle of the lateral sides with the triangle base is 30 degrees.
What is the volume of the pyramid?
12StartRoot 3 EndRoot cm3
16StartRoot 3 EndRoot cm3
24StartRoot 3 EndRoot cm3
32StartRoot 3 EndRoot cm3
All Answers 1
Answered by
GPT-5 mini
AI
Answer: 16 cm^3.
Work:
- Base side a = 4√3 so circumradius R = a/√3 = 4.
- If each lateral edge makes 30° with the base plane, then for a lateral edge of length l and height h we have h = l sin30 = l/2, and l^2 = h^2 + R^2. Combining gives (2h)^2 = h^2 + R^2 ⇒ 3h^2 = R^2 ⇒ h = R/√3 = 4/√3 = (4√3)/3.
- Volume V = (1/3)(base area)·h = (1/3)(12√3)·(4√3/3) = 16 cm^3.
(Note: none of the listed multiple-choice answers equals 16 cm^3 as written; the intended correct choice is the one corresponding to 16 cm^3.)
Work:
- Base side a = 4√3 so circumradius R = a/√3 = 4.
- If each lateral edge makes 30° with the base plane, then for a lateral edge of length l and height h we have h = l sin30 = l/2, and l^2 = h^2 + R^2. Combining gives (2h)^2 = h^2 + R^2 ⇒ 3h^2 = R^2 ⇒ h = R/√3 = 4/√3 = (4√3)/3.
- Volume V = (1/3)(base area)·h = (1/3)(12√3)·(4√3/3) = 16 cm^3.
(Note: none of the listed multiple-choice answers equals 16 cm^3 as written; the intended correct choice is the one corresponding to 16 cm^3.)
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