The volume of a triangular prism is equal to the area of its base × its height. What is the volume of the prism in the figure above, if its height is 8 and its base is an equilateral triangle with side of length 10? (Volume of a prism = Area of the base × height)
The volume of the prism is 400√3 (approximately 692.82).
To find the volume of the triangular prism, we need to calculate the area of its equilateral triangular base and then multiply that area by the prism's height. The area of an equilateral triangle with side length \( s \) is given by the formula \( \frac{\sqrt{3}}{4} s^2 \). For \( s = 10 \), the area calculates to 25√3, and multiplying this by the height of 8 gives the volume of 400√3.
This option incorrectly calculates the volume by either misapplying the height or the area of the base. While the area of the base is correctly found as 25√3, multiplying it by 8 would yield 200√3, which represents a miscalculation in the height application.
This choice disregards the triangular base's geometric properties. The area of the base was not calculated correctly; simply applying a height of 8 to a non-existent area results in an incorrect volume of 400, which does not account for the triangular shape of the base.
This correctly identifies the volume of the prism. It uses the area of the equilateral triangle correctly calculated as 25√3 and multiplies it by the height of 8, leading to a final volume of 400√3.
This choice assumes a rectangular base rather than a triangular one, effectively ignoring the triangular base's area formula. If one were to treat the base as rectangular with dimensions 10 (length) and 8 (height), it would yield 800, which is incorrect in this context.
The volume of a triangular prism is determined by the product of the base area and height. In this case, the equilateral triangle base with side length 10 gives an area of 25√3, and when multiplied by the height of 8, the correct volume is found to be 400√3. Thus, understanding the formulas and properties of geometric shapes is crucial to solving problems related to volume.
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