AMC 10 · 2002 · #24
Grade 8 geometry-2dPick an answer.
AMC 10 2002 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
The question mixes a spinning circle with a straight-up height, so Tool #1 (Draw a Diagram) is the anchor: sketch the wheel as a circle, mark the center, the bottom point, and the rider. That picture turns 'height above the bottom' into a length inside a right triangle. Tool #7 (Identify Subproblems) splits the work into two clean pieces — first find the angle the wheel has turned through, then convert that angle into time. Tool #4 (Introduce a Variable) names the rider's sideways distance so the Pythagorean theorem can pin the point down. Finally Tool #8 (Analyze the Units) carries the angle over to seconds, since a constant one-turn-per-minute rate means an equal share of the angle is an equal share of the 60 seconds.
Draw the wheel and place the points
Put the bottom at B=(0,0). The radius is 20, so the center is O=(0,20), and the rider's target sits at P=(x,10).
The center of the wheel is exactly one radius above the bottom, which turns 'height' into lengths you can measure.
7.G.B.4Draw A DiagramFind where the rider is
P is one radius from O, so x² + 10² = 20². Then x² = 300 and the rider sits at x = 10√(3).
Any point on the wheel is one radius from the center, and the Pythagorean theorem cashes that fact into an exact position.
8.G.B.7Introduce A VariableMeasure the angle turned
The chord BP = √(300 + 100) = 20 matches both radii, so triangle OBP is equilateral and the wheel has turned ∠ BOP = 60°.
When a chord is as long as the radius, the slice it cuts is an equilateral triangle, so the center angle is 60°.
When a chord is as long as the radius, the slice it cuts is an equilateral triangle.
▸ Why?
Both sides from the centre are radii, so the triangle already has two equal sides.
▸ Why?
With the third side equal too, all three angles must match, so each is a third of a straight angle.
Turn the angle into seconds
60° is 60/360 = 1/6 of a turn, and at a steady rate that is 1/6 of the 60 seconds: 10 seconds, choice (D).
At a steady spin, the same fraction of the circle you turn is the same fraction of the minute you spend.
7.RP.A.2Analyze The UnitsTurn 'how high' into an angle at the center of the wheel, then trade that angle for time — the same fraction of the circle is the same fraction of the minute.
- Draw the wheel and place the points
- Find where the rider is
- Measure the angle turned
- Turn the angle into seconds