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Maximizing Volume

Maximizing Volume
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2018-12-20 02:45
TriangleFunctionSolid GeometryCalculusOptimization
This example demonstrates how to maximize the volume of a triangular prism under a given perimeter constraint by adjusting the base length and height of the triangle. Key math concepts include volume calculation, means inequality, exponential functions, triangle properties, and equation solving. GeoGebra commands and objects used in construction are Sequence, Point, Polygon, Rotate, Prism, LaTeX, TableText, and Vector. Suitable for high school solid geometry and calculus instruction, focusing on optimization problems.

GeoGebra Commands & Parameters

  1. Sequence((i, 0), i, x(Box1Anchor), x(Box1Anchor) + pLength / 2, 0.1) → xPoints
  2. Point(xPoints) → X
  3. Polygon(Box1Anchor, Box1Anchor + (0, pLength), Box1Anchor + (pWidth, pLength), Box1Anchor + (pWidth, 0)) → paper, f, g, h, i
  4. Polygon(Box1Anchor, Box1Anchor + (boxWidth, 0), 4) → Box1, j, k, l, m, A, B
  5. Polygon(Box2Anchor, B2, 4) → Box2, n, p, q, r, C, D
  6. Polygon(Box3Anchor, Box3Anchor + (boxWidth, 0), 4) → Box3, s, t, a, b, E, F
  7. Polygon(Box4Anchor, Box4Anchor + (boxWidth, 0), 4) → Box4, c, d, e, f_1, G, H
  8. Rotate(T1, 90°, B1, yAxis) → T1'
  9. Polygon(B1, B2, B3, B4) → pBase, b1, b2, b3, b4
  10. Prism(pBase, T1') → o, I, edge2, edge3, edge4, edge5, edge6, edge7, edge8, J, K, face1, face2, face3, face4, face5, edge1
  11. LaTeX("X") → xLbl
  12. LaTeX(prismLength) → B6
  13. LaTeX(prismWidth) → B7
  14. LaTeX(prismHeight) → B8
  15. LaTeX(prismVolume) → B9
  16. LaTeX(x(X)) → B5
  17. TableText({A5:A9, B5:B9}, "hr |_") → results
  18. LaTeX("\cellcolor{#ffcc80}\text{\phantom{abcd}X}") → A5
  19. Vector((0, 0, 0), (14, 0, 0)) → xAxisArrow
  20. Vector((0, 0, 0), (0, 12.5, 0)) → yAxisArrow
  21. LaTeX((((("\text{\textbf{Question: }" * MainInstruc) * "\\\textbf{Instructions: }") * SpecInstruc) * "}")) → allInstruc