Tutorial
Real Numbers – Episode 10, The Art Thief at the Window
- New Problem Definition: The episode introduces a challenge to determine the maximum height of a rectangular framed painting that can be slid through a specific window opening.
- Window Geometry: The window consists of a square with a side length of one meter, capped by a semicircle on top where the semicircle's diameter aligns with the top edge of the square.
- Painting Constraints: The paintings are rigid two-dimensional rectangles with a uniform width of three meters and varying heights, which cannot be bent or folded during transport.
- Geometric Challenge Resolution: The calculated optimal aiming point to return the toy boat "Wavy" to the friend's dock is 127.5 meters (255/2 meters) upstream from the friend.
- Velocity Ratio: The boat's motor speed is established as $5/3$ times the speed of the river current, derived from a 100-meter crossing distance and a 60-meter downstream drift.
- Algebraic Solution Method: Solving the quadratic equation $16x^2 - 1080x - 122,400 = 0$ yields the primary solution of $x = 127.5$ and a discarded negative root of $-60$.
- Geometric Verification: The algebraic result was independently verified using a purely geometric approach involving similar triangles ($ACE$ and $BDE$) and the tangent double-angle formula ($\tan(2\theta) = 15/8$).
- Geometric Insight: The analysis revealed that the boat's trajectory creates an isosceles triangle ($ADE$) and a specific angular relationship where the aiming angle is exactly double the angle of the river current's drift relative to the cross-stream direction.
- Submission Instructions: Viewers are invited to submit solutions for the art museum window problem, including photos of their work, on the show's website.