$\"\"$

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The base of a solid $\"\"$ is the triangular region with vertices $\"\"$ , $\"\"$ and $\"\"$ .

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Cross sections perpendicular to the $\"\"$ -axis are equalteral triangles.

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Draw the top view of the solid with vertices $\"\"$ , $\"\"$ , $\"\"$ and cross sections perpendicular $\"\"$ -axis with side length as $\"\"$ .

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$\"\"$

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Observe the figure,

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Find the line equation of side of the triangle.

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Point-slope form of line equation: $\"\"$ .

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Substitute $\"\"$ and $\"\"$ in above formula.

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$\"\"$

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$\"\"$

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Area of the equalateral triangle is $\"\"$ .

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Substitute $\"\"$ in $\"\"$ .

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$\"\"$ .

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$\"\"$

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Find the volume of the solid by integrating the area with respect to $\"\"$ over the limits $\"\"$ to $\"\"$ .

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$\"\"$

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$\"\"$

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$\"\"$

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$\"\"$

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Volume of the solid is $\"\"$ .

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$\"\"$

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Volume of the solid is $\"\"$ .