12. 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)
Answer: C
The volume of the prism is 400√3 (approximately 692.82).
To find the volume of the triangular prism, we calculate the area of the base, which is an equilateral triangle with a side length of 10, and then multiply it by the height of 8. The area of the equilateral triangle is (√3/4) × side², leading to a volume of 400√3 when calculated with the given height.
A) 200√3(approximately 346.41)
This option is incorrect because the calculation for the area of the equilateral triangle does not yield an area that, when multiplied by the height, results in 200√3. The area of the triangle alone exceeds this value, making this option invalid.
B) 400
This option is incorrect as it does not account for the triangular base's area. The volume must incorporate the area of the triangular base, which is greater than 400 when multiplied by the height of 8.
C) 400√3(approximately 692.82)
This option is correct. The area of the base (the equilateral triangle) is calculated as (√3/4) × (10)² = 25√3. Multiplying this area by the height of 8 results in a volume of 200√3 × 2 = 400√3.
D) 800
This option is incorrect as it fails to consider the geometry of the base. The volume cannot simply equal the height multiplied by a constant without calculating the area of the triangular base first.
Conclusion
The correct answer, 400√3, is derived from accurately calculating the area of the equilateral triangle base and multiplying it by the prism's height. All other options either miscalculate the area or ignore the triangular shape of the base, leading to incorrect volume results.