Volume of a solid with equilateral triangle cross-sections

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SUMMARY

The volume of a solid with equilateral triangle cross-sections, bounded by the parabola x² = 8y and the line y = 4, is calculated using integration. The correct formula for the area of the equilateral triangle is A = (s²√3)/4, where s is the side length derived from the equation x = √(8y). The volume is determined by integrating the area from y = 0 to y = 4, resulting in a final volume of 64√3. The initial miscalculation stemmed from an incorrect expression for x, which should be √(8y) instead of (8y)^(1/2).

PREREQUISITES
  • Understanding of calculus, specifically integration techniques
  • Familiarity with the properties of equilateral triangles
  • Knowledge of parabolic equations and their graphical representations
  • Ability to manipulate algebraic expressions involving square roots
NEXT STEPS
  • Study integration techniques for calculating volumes of solids of revolution
  • Learn about the properties and formulas related to equilateral triangles
  • Explore the graphical representation of parabolas and their intersections with linear equations
  • Practice solving similar problems involving cross-sections of solids
USEFUL FOR

Students and educators in mathematics, particularly those focusing on calculus and geometry, as well as anyone interested in solving volume-related problems involving cross-sections.

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The base of a solid is the region bounded by the parabola x2 = 8y and the line y = 4 and each plane section perpendicular to the y-axis is an equilateral triangle. What is the volume of the solid?(Barron's Problem)

so I solved for x since y must be used because the cross section is perpendicular to the y-axis. x=(8y)1/2. The area equation for an equilateral triangle is A = s2√(3)/4
So I figured the volume of the cross section would be equal to the integral from 0 to 4 of (8y√(3))/4 which gives me 16√(3) which is apparently wrong. The answer is supposed to be 64√(3). I don't know what I did wrong.
 
Last edited:
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82= 64, not 8!
 
HallsofIvy said:
82= 64, not 8!

I wrote down solving for x incorrectly its supposed to be √(8y) which allows my confusion to continue.
 

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