Height as a Function of Volume

Loading the math board and drawing, please wait…

2017-07-16 23:26

This example demonstrates the dynamic relationship between the height and volume of a cylinder. Core math concepts include the volume formula of a cylinder, indefinite integral, statistical characteristics, coordinate geometry, and trigonometry. It is constructed using GeoGebra commands like If, Polygon, Point, Translate, Sequence, Textfield, Ray, Line, and Intersect, enabling interactive visualization of volume-height changes. Suitable for high school and above mathematics education, focusing on solid geometry and calculus applications.

GeoGebra Commands & Parameters

If(Vmax > 49, (round(Vmax / 100) * 10), 5) → unt
Polygon((r, 0), (r, h), ((-r), h), ((-r), 0)) → poly1, f, g, i, j
Polygon((r, 0), (r, fillh), ((-r), fillh), ((-r), 0)) → poly2, k, l, m, n
Point(Ray[(0, 0), (1, 0)]) → R
Point(Ray[(0, 0), (0, 1)]) → H
Translate(c, (0, fillh)) → c'
Translate(c, (0, h)) → c'_1
Sequence(If[(-r) ≤ x ≤ (-0.6 * r), q(x) + (i * unth)], i, 1, untn) → list3
Textfield(h) → InputBox1
Textfield(r) → InputBox2
Ray((0, fillh), (0, fillh + 0.1)) → e
Line(A, xAxis) → f_1
Intersect(f_1, i) → B

AI Drawing Prompt Reference

Draw a vertical cross-section of a cylinder with adjustable radius and height via input boxes. The section is a rectangle, with the lower part filled to represent liquid level, whose height changes with volume. Place two draggable points on the axes to control radius and height. Add horizontal guide lines at the top and at the fill level, plus a short vertical segment at the fill level. Mark a point where the left side of the rectangle meets a horizontal line, indicating the height-volume relationship. Show the dynamic change of height with volume.

The prompt above was compiled from the original drawing commands and is for reference only.