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Add the circular string art problem

Vectornaut 2025-05-27 03:43:25 +00:00
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Test-problems.md

@ -139,4 +139,20 @@ These problems impose various kinds of *soft constraints* on top of the *hard co
#### Statement
“David Seppala-Holtzman of St. Joseph’s College New York gives us the following problem. A customer orders five identical perfectly spherical cannonballs from Adderley’s Cannonball Emporium, and it’s your job to pack them for shipping. The Emporium ships only in rectangular boxes but can construct such boxes with any desired dimensions. You have a choice of packing the cannonballs so that their centers form a square pyramid or two triangular pyramids, as in figure 1 [see source], but you can orient the arrangements however you like inside the box. The shipping cost will be proportional to the sum of the length, width, and height of the box. In **Frugal Firepower**, determine which arrangement allows you to minimize the shipping cost.”
“David Seppala-Holtzman of St. Joseph’s College New York gives us the following problem. A customer orders five identical perfectly spherical cannonballs from Adderley’s Cannonball Emporium, and it’s your job to pack them for shipping. The Emporium ships only in rectangular boxes but can construct such boxes with any desired dimensions. You have a choice of packing the cannonballs so that their centers form a square pyramid or two triangular pyramids, as in figure 1 [see source], but you can orient the arrangements however you like inside the box. The shipping cost will be proportional to the sum of the length, width, and height of the box. In **Frugal Firepower**, determine which arrangement allows you to minimize the shipping cost.”
## Algebraic relations
### Circular string art
#### Statement
Draw a pair of perpendicular lines, which we'll treat as coordinate axes. Draw a unit circle in the positive quadrant which is tangent to both axes. Draw a line which is tangent to the circle, letting $s$ and $t$ be the coordinates where it crosses the axes. Find the relationship between $s$ and $t$.
#### Solution
$$s + t = 1 + \tfrac{1}{2} st$$
#### Note
[Geometry Expressions](https://geometryexpressions.com) can solve this problem handily!