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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․ Найдите объел пиралиды.

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Q. Высота правильной треугольной пиралиды равна 43 4 \sqrt{3} , а боковая грань образует с плоскостью основания угол 6060․ Найдите объел пиралиды.
  1. Calculate apothem length: Calculate the length of the apothem (slant height) of the pyramid using the height and the 6060-degree angle.\newlineapothem =heightcos⁡(60°)= \frac{\text{height}}{\cos(60°)}\newline=43(1/2)= \frac{4\sqrt{3}}{(1/2)}\newline=83= 8\sqrt{3}
  2. Find base side length: Find the side length of the triangular base using the apothem and the 6060-degree angle.\newlineside=apothem×sin⁡(60∘)\text{side} = \text{apothem} \times \sin(60^\circ)\newline=83×(3/2)= 8\sqrt{3} \times (\sqrt{3}/2)\newline=12= 12
  3. Calculate base area: Calculate the area of the triangular base using the side length.\newlineArea = (3/4)×side2(\sqrt{3}/4) \times \text{side}^2\newline= (3/4)×122(\sqrt{3}/4) \times 12^2\newline= (3/4)×144(\sqrt{3}/4) \times 144\newline= 36336\sqrt{3}
  4. Find pyramid volume: Find the volume of the pyramid using the area of the base and the height.\newlineVolume = (13)×base_area×height(\frac{1}{3}) \times \text{base\_area} \times \text{height}\newline= (13)×363×43(\frac{1}{3}) \times 36\sqrt{3} \times 4\sqrt{3}\newline= 144×3144 \times 3\newline= 432432

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