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Высота правильной треугольной пиралиды равна 4sqrt3, а боковая грань образует с плоскостью основания угол 60". Найдите объел пиралиды.

Высота правильной треугольной пиралиды равна 43 4 \sqrt{3} , а боковая грань образует с плоскостью основания угол 6060

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Q. Высота правильной треугольной пиралиды равна 43 4 \sqrt{3} , а боковая грань образует с плоскостью основания угол 6060
  1. Calculate Base Side Length: Calculate the length of the side of the base triangle using the height and the angle given.\newlineSince the height 434\sqrt{3} and the angle 6060 degrees between the side and the base are given, we can use trigonometry to find the side of the base triangle.\newlineLet's call the side of the base triangle ′a′'a'.\newlineWe know that tan⁡(60)=heighta/2\tan(60) = \frac{\text{height}}{a/2}, so a=2×heighttan⁡(60)a = 2 \times \frac{\text{height}}{\tan(60)}.\newlinea=2×433=8a = 2 \times \frac{4\sqrt{3}}{\sqrt{3}} = 8.
  2. Calculate Base Triangle Area: Calculate the area of the base triangle.\newlineThe area of an equilateral triangle is (3/4)×a2(\sqrt{3}/4) \times a^2.\newlineArea = (3/4)×82=23×64/4=23×16=323(\sqrt{3}/4) \times 8^2 = 2\sqrt{3} \times 64 / 4 = 2\sqrt{3} \times 16 = 32\sqrt{3}.
  3. Calculate Pyramid Volume: Calculate the volume of the pyramid.\newlineThe volume of a pyramid is (13)×base area×height(\frac{1}{3}) \times \text{base area} \times \text{height}.\newlineVolume = (13)×323×43=(13)×128×3=128(\frac{1}{3}) \times 32\sqrt{3} \times 4\sqrt{3} = (\frac{1}{3}) \times 128 \times 3 = 128.

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