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Sugar Cube Company: Volume Comparison of Different Sizes

Sugar Cube Company: Volume Comparison of Different Sizes
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2015-05-15 02:41
Solid GeometryGeometric Construction
This example demonstrates the volume comparison of sugar cubes in different sizes. Key mathematical concepts include cube volume calculation and proportionality. It uses Corner, Cube, TableText, and other GeoGebra commands for construction. Suitable for high school students to learn cube volume calculation and proportionality.

GeoGebra Commands & Parameters

  1. LaTeX(steps) → counter
  2. LaTeX("\text{Use the buttons below to return to the original problem or}\cr\text{randomly generate a similar practice problem.}") → RandomizationText
  3. Corner(1) → A
  4. Corner(3) → B
  5. Cube(((-hs), (-hs), (-hs)), (hs, (-hs), (-hs)), xOyPlane) → b, C, face4, faceCDHG, faceEFGH, edge2, edge1, edge4, edgeCD, edge3, edgeDH, edgeEH, D, edgeEF, edge5, edgeFG, edgeCG, edgeGH, E, F, G, H, face1, face3, face2
  6. Sphere((0, 0, 0), 0.62) → a
  7. LaTeX((((("\text{small cubes: side length of }\textcolor{#000099}{" * s_s) * "}\text{ cm}\cr\text{large cubes: side length of }\textcolor{#000099}{") * s_l) * "}\text{ cm}")) → Problem
  8. LaTeX("\text{Move the slider to solve the problem.}") → text0
  9. LaTeX("\text{Find a formula the company can use to determine the radius of a sugar sphere with}\cr\text{the same volume as a sugar cube of side length }\textcolor{#000099}{\mathit{x}}.") → text1
  10. LaTeX("\text{Recall the formula for the volume, }\textcolor{#000099}{\mathit{V}_{sphere}}\text{, of a sphere of radius }\textcolor{#000099}{\mathit{r}}.") → text2
  11. LaTeX("\textcolor{#000099}{\mathit{V}_{sphere}=\frac{4}{3}\pi \mathit{r}^3}") → text3
  12. LaTeX("\text{Recall the formula for the volume, }\textcolor{#000099}{\mathit{V}_{cube}}\text{, of a cube of side length }\textcolor{#000099}{\mathit{x}}.") → text4
  13. LaTeX("\textcolor{#000099}{\mathit{V}_{cube}= \mathit{x}^3}") → text5
  14. LaTeX("\text{To find the radius of a sphere with the same volume as a cube of side length }\textcolor{#000099}{\mathit{x}},\cr\text{set the volume formulas equal, and solve for the radius }\textcolor{#000099}{\mathit{r}}.") → text6
  15. TableText({{LaTeX["\textcolor{#000099}{\mathit{V}_{sphere} = \mathit{V}_{cube}}"], LaTeX["\text{Set the volumes of a sphere and a cube equal to each other.}"]}}) → text7
  16. TableText({{LaTeX["\textcolor{#000099}{\mathit{V}_{sphere} = \mathit{V}_{cube}}"], LaTeX["\text{Set the volumes of a sphere and a cube equal to each other.}"]}, {LaTeX["\textcolor{#000099}{\left(\frac{4}{3}\pi \mathit{r}^3\right)=\left(\mathit{x}^3\right)}"], LaTeX["\text{Substitute }\textcolor{#000099}{\frac{4}{3}\pi \mathit{r}^3}\text{ for }\textcolor{#000099}{\mathit{V}_{sphere}}\text{ and }\textcolor{#000099}{\mathit{x}^3}\text{ for }\textcolor{#000099}{\mathit{V}_{cube}}."]}}) → text8
  17. TableText({{LaTeX["\textcolor{#000099}{\mathit{V}_{sphere} = \mathit{V}_{cube}}"], LaTeX["\text{Set the volumes of a sphere and a cube equal to each other.}"]}, {LaTeX["\textcolor{#000099}{\left(\frac{4}{3}\pi \mathit{r}^3\right)=\left(\mathit{x}^3\right)}"], LaTeX["\text{Substitute }\textcolor{#000099}{\frac{4}{3}\pi \mathit{r}^3}\text{ for }\textcolor{#000099}{\mathit{V}_{sphere}}\text{ and }\textcolor{#000099}{\mathit{x}^3}\text{ for }\textcolor{#000099}{\mathit{V}_{cube}}."]}, {LaTeX["\textcolor{#000099}{\frac{3}{4\pi}\cdot\frac{4}{3}\pi \mathit{r}^3=\frac{3}{4\pi}\cdot\mathit{x}^3}"], LaTeX["\text{Multiply both sides by }\textcolor{#000099}{\frac{3}{4\pi}}\text{, the reciprocal of }\textcolor{#000099}{\frac{4\pi}{3}}\text{, to isolate }\textcolor{#000099}{\mathit{r}^3}."]}}) → text9
  18. TableText({{LaTeX["\textcolor{#000099}{\mathit{V}_{sphere} = \mathit{V}_{cube}}"], LaTeX["\text{Set the volumes of a sphere and a cube equal to each other.}"]}, {LaTeX["\textcolor{#000099}{\left(\frac{4}{3}\pi \mathit{r}^3\right)=\left(\mathit{x}^3\right)}"], LaTeX["\text{Substitute }\textcolor{#000099}{\frac{4}{3}\pi \mathit{r}^3}\text{ for }\textcolor{#000099}{\mathit{V}_{sphere}}\text{ and }\textcolor{#000099}{\mathit{x}^3}\text{ for }\textcolor{#000099}{\mathit{V}_{cube}}."]}, {LaTeX["\textcolor{#000099}{\frac{3}{4\pi}\cdot\frac{4}{3}\pi \mathit{r}^3=\frac{3}{4\pi}\cdot\mathit{x}^3}"], LaTeX["\text{Multiply both sides by }\textcolor{#000099}{\frac{3}{4\pi}}\text{, the reciprocal of }\textcolor{#000099}{\frac{4\pi}{3}}\text{, to isolate }\textcolor{#000099}{\mathit{r}^3}."]}, {LaTeX["\textcolor{#000099}{\mathit{r}^3=\frac{3}{4\pi}\mathit{x}^3}"], LaTeX["\text{Simplify.}"]}}) → text10
  19. TableText({{LaTeX["\textcolor{#000099}{\mathit{V}_{sphere} = \mathit{V}_{cube}}"], LaTeX["\text{Set the volumes of a sphere and a cube equal to each other.}"]}, {LaTeX["\textcolor{#000099}{\left(\frac{4}{3}\pi \mathit{r}^3\right)=\left(\mathit{x}^3\right)}"], LaTeX["\text{Substitute }\textcolor{#000099}{\frac{4}{3}\pi \mathit{r}^3}\text{ for }\textcolor{#000099}{\mathit{V}_{sphere}}\text{ and }\textcolor{#000099}{\mathit{x}^3}\text{ for }\textcolor{#000099}{\mathit{V}_{cube}}."]}, {LaTeX["\textcolor{#000099}{\frac{3}{4\pi}\cdot\frac{4}{3}\pi \mathit{r}^3=\frac{3}{4\pi}\cdot\mathit{x}^3}"], LaTeX["\text{Multiply both sides by }\textcolor{#000099}{\frac{3}{4\pi}}\text{, the reciprocal of }\textcolor{#000099}{\frac{4\pi}{3}}\text{, to isolate }\textcolor{#000099}{\mathit{r}^3}."]}, {LaTeX["\textcolor{#000099}{\mathit{r}^3=\frac{3}{4\pi}\mathit{x}^3}"], LaTeX["\text{Simplify.}"]}, {LaTeX["\textcolor{#000099}{\sqrt[3]{\mathit{r}^3}=\sqrt[3]{\frac{3}{4\pi}\mathit{x}^3}}"], LaTeX["\text{Take the cube root of both sides.}"]}}) → text11
  20. TableText({{LaTeX["\textcolor{#000099}{\mathit{V}_{sphere} = \mathit{V}_{cube}}"], LaTeX["\text{Set the volumes of a sphere and a cube equal to each other.}"]}, {LaTeX["\textcolor{#000099}{\left(\frac{4}{3}\pi \mathit{r}^3\right)=\left(\mathit{x}^3\right)}"], LaTeX["\text{Substitute }\textcolor{#000099}{\frac{4}{3}\pi \mathit{r}^3}\text{ for }\textcolor{#000099}{\mathit{V}_{sphere}}\text{ and }\textcolor{#000099}{\mathit{x}^3}\text{ for }\textcolor{#000099}{\mathit{V}_{cube}}."]}, {LaTeX["\textcolor{#000099}{\frac{3}{4\pi}\cdot\frac{4}{3}\pi \mathit{r}^3=\frac{3}{4\pi}\cdot\mathit{x}^3}"], LaTeX["\text{Multiply both sides by }\textcolor{#000099}{\frac{3}{4\pi}}\text{, the reciprocal of }\textcolor{#000099}{\frac{4\pi}{3}}\text{, to isolate }\textcolor{#000099}{\mathit{r}^3}."]}, {LaTeX["\textcolor{#000099}{\mathit{r}^3=\frac{3}{4\pi}\mathit{x}^3}"], LaTeX["\text{Simplify.}"]}, {LaTeX["\textcolor{#000099}{\sqrt[3]{\mathit{r}^3}=\sqrt[3]{\frac{3}{4\pi}\mathit{x}^3}}"], LaTeX["\text{Take the cube root of both sides.}"]}, {LaTeX["\textcolor{#000099}{\sqrt[3]{\mathit{r}^3}=\sqrt[3]{\frac{3}{4\pi}}\sqrt[3]{\mathit{x}^3}}"], LaTeX["\text{Simplify the right side into a product of cube roots.}"]}}) → text12
  21. TableText({{LaTeX["\textcolor{#000099}{\mathit{V}_{sphere} = \mathit{V}_{cube}}"], LaTeX["\text{Set the volumes of a sphere and a cube equal to each other.}"]}, {LaTeX["\textcolor{#000099}{\left(\frac{4}{3}\pi \mathit{r}^3\right)=\left(\mathit{x}^3\right)}"], LaTeX["\text{Substitute }\textcolor{#000099}{\frac{4}{3}\pi \mathit{r}^3}\text{ for }\textcolor{#000099}{\mathit{V}_{sphere}}\text{ and }\textcolor{#000099}{\mathit{x}^3}\text{ for }\textcolor{#000099}{\mathit{V}_{cube}}."]}, {LaTeX["\textcolor{#000099}{\frac{3}{4\pi}\cdot\frac{4}{3}\pi \mathit{r}^3=\frac{3}{4\pi}\cdot\mathit{x}^3}"], LaTeX["\text{Multiply both sides by }\textcolor{#000099}{\frac{3}{4\pi}}\text{, the reciprocal of }\textcolor{#000099}{\frac{4\pi}{3}}\text{, to isolate }\textcolor{#000099}{\mathit{r}^3}."]}, {LaTeX["\textcolor{#000099}{\mathit{r}^3=\frac{3}{4\pi}\mathit{x}^3}"], LaTeX["\text{Simplify.}"]}, {LaTeX["\textcolor{#000099}{\sqrt[3]{\mathit{r}^3}=\sqrt[3]{\frac{3}{4\pi}\mathit{x}^3}}"], LaTeX["\text{Take the cube root of both sides.}"]}, {LaTeX["\textcolor{#000099}{\sqrt[3]{\mathit{r}^3}=\sqrt[3]{\frac{3}{4\pi}}\sqrt[3]{\mathit{x}^3}}"], LaTeX["\text{Simplify the right side into a product of cube roots.}"]}, {LaTeX["\textcolor{#000099}{\mathit{r}=\left(\sqrt[3]{\frac{3}{4\pi}}\right) \mathit{x}}"], LaTeX["\text{Simplify.}"]}}) → text13
  22. LaTeX("\text{Thus, a sphere with radius }\textcolor{#000099}{\mathit{r}=\left(\sqrt[3]{\frac{3}{4\pi}}\right)\mathit{x}}\text{ has the same volume as a cube with side length }\textcolor{#000099}{\mathit{x}}.") → text14
  23. LaTeX("\text{Use the formula }\textcolor{#000099}{\mathit{r}=\left(\sqrt[3]{\frac{3}{4\pi}}\right)\mathit{x}}\text{ to determine the radii required for the proposed sugar spheres.}") → text15
  24. TableText({{LaTeX["\text{}"], LaTeX["\text{Side of cube in cm}"], LaTeX["\text{Radius of sphere in cm}"]}, {LaTeX["\text{Small}"], LaTeX[(("\textcolor{#000099}{" * s_s) * "}")], LaTeX[""]}, {LaTeX["\text{Large}"], LaTeX[(("\textcolor{#000099}{" * s_l) * "}")], LaTeX[""]}}, "c|_") → text16
  25. TableText({{LaTeX["\text{}"], LaTeX["\text{Side of cube in cm}"], LaTeX["\text{Radius of sphere in cm}"]}, {LaTeX["\text{Small}"], LaTeX[(("\textcolor{#000099}{" * s_s) * "}")], LaTeX[(("\textcolor{#000099}{\mathit{r}=\left(\sqrt[3]{\frac{3}{4\pi}}\right)\left(" * s_s) * "\right)}")]}, {LaTeX["\text{Large}"], LaTeX[(("\textcolor{#000099}{" * s_l) * "}")], LaTeX[(("\textcolor{#000099}{\mathit{r}=\left(\sqrt[3]{\frac{3}{4\pi}}\right)\left(" * s_l) * "\right)}")]}}, "c|_") → text17
  26. TableText({{LaTeX["\text{}"], LaTeX["\text{Side of cube in cm}"], LaTeX["\text{Radius of sphere in cm}"]}, {LaTeX["\text{Small}"], LaTeX[(("\textcolor{#000099}{" * s_s) * "}")], LaTeX[(((("\textcolor{#000099}{\mathit{r}=\left(\sqrt[3]{\frac{3}{4\pi}}\right)\left(" * s_s) * "\right)\approx") * r_s) * "}")]}, {LaTeX["\text{Large}"], LaTeX[(("\textcolor{#000099}{" * s_l) * "}")], LaTeX[(((("\textcolor{#000099}{\mathit{r}=\left(\sqrt[3]{\frac{3}{4\pi}}\right)\left(" * s_l) * "\right)\approx") * r_l) * "}")]}}, "c|_") → text18