When Does Height Equal Zero?

Algebra 2 · Axiom Academy

A Ferris wheel ride shows why a trig equation can have zero, one, or infinitely many solutions. You're riding a wheel with a 25 ft radius whose center sits 30 ft above the ground — as it turns, does your height ever reach 0 ? Hit ▶︎ Go and ride two full turns — watch the height swing between the top and the bottom of the wheel. Drag the target line anywhere from the ground to the top — some heights get hit twice a lap, one just grazes the very top or bottom, and some are never reached at all. Fix the target at the wheel's lowest point (5 ft — as close to the ground as it gets) and ride however many rotations you like. The bottom always arrives right on schedule. Same idea every time a periodic model needs solving: check whether the target is even reachable — is it inside the model's range? — then use angle = base angle + period · n to sweep out every solution. It's the trick behind tides, AC voltage, and any cycle that repeats.

This is the written version of the interactive lesson above. See the full Algebra 2 course.