Geometry Puzzle – Challenge 70
A cubical wood block weighs 5 kg. How much does a cube of the same wood weigh if its sides are three times as long? Weight is proportional to volume, and cube volume scales by the cube of the linear factor. So the new cube has 3^3 = 27 times the volume, and 27 times the weight: 27 times 5 = 135 kg.
Key takeaways:
- Weight (of uniform material) is proportional to volume.
- Cube volume = (side)^3.
- Scaling each side by factor k scales volume by k^3.
- Tripling the side multiplies volume (and weight) by 27.
- New weight = 27 times 5 = 135 kg.
A great Math puzzle to help students develop critical thinking and logic skills in a fun way. Let’s challenge your brain! For additional educational resources
Challenge:
A cubical block of wood weighs 5 kilograms. How much will another cube of the same wood weigh if its sides are three times as long?
A- 15
B- 30
C- 60
D- 105
E- 135
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The correct answer is E.
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Weight is proportional to volume.
Volume of a cube = (one side)\(^3\)
Let x be the side of the first cube. So:
Volume \(= x^3\)
The sides of the second cube are three times as long. So:
Volume \(= (3x)^3 = 27x^3\)
The ratio of the volumes of the old and new cubes will be 1: 27. So, if the first weighs 5 kilograms, the second weighs 27 × 5 kilograms =135 kilogram.The Best Books to Ace Algebra
Frequently Asked Questions
Is the volume scale factor the length scale factor cubed?
Yes. If every length of a solid is multiplied by k, its volume is multiplied by k³, because volume uses three dimensions. In this puzzle the cube’s sides are tripled, so the volume grows by 3³ = 27. Since the wood is the same, the weight also grows by 27: 5 kg × 27 = 135 kg, not 15 kg.
How do you find the volume scale factor?
Cube the length scale factor, or divide the new volume by the old one. For two similar shapes with sides 2 cm and 6 cm, the length scale factor is 6 ÷ 2 = 3, so the volume scale factor is 3³ = 27. Going the other way, if one cube weighs 135 kg and the other 5 kg, 135 ÷ 5 = 27.
What is the scale factor volume formula?
New volume = k³ × original volume, where k is the length scale factor. Area follows k², so surface area changes by a squared factor. For the cubes in this puzzle, k = 3, so surface area grows 9 times while volume and weight grow 27 times. If k = 1/2, the volume shrinks to 1/8 of the original.
How do you find the scale factor of a cube?
Divide a side of the new cube by the matching side of the original. If you only know volumes (or weights, for the same material), divide them and take the cube root. Here, 135 kg ÷ 5 kg = 27, and ∛27 = 3, so the larger cube’s sides are three times as long, which matches the puzzle.
Related Lessons You May Like
- How to find the area of rectangles
- How to find the perimeter of rectangles
- How to solve angles and angle measure
- How to solve percent of change
- How to find volume and surface area of cubes
If your student enjoys puzzles like this, Geometry for Beginners works the same kinds of relationships inside a full curriculum. Pre-Algebra for Beginners covers the algebraic foundations.
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