AMC 10 · 2011 · #12

Grade 7 rate-ratiogeometry-2d
Archetype Coordinate Plane Figure Computation └ Sub-archetype Figure Constraint Equations for Unknown Quantity
path-length-comparisonratelinear-equations-one-var convert-to-algebra ↑ Prerequisites: rate
📏 Medium solution 💡 2 insights
Problem
A running track has two straight sides and two semicircular ends, like a stadium. The track is 6 meters wide. Walking around the outer edge takes 36 seconds longer than walking around the inner edge, at the same steady speed. Find that speed in meters per second.

Pick an answer.

(A)
$\frac{\pi}{3}$
(B)
$\frac{2\pi}{3}$
(C)
$\pi$
(D)
$\frac{4\pi}{3}$
(E)
$\frac{5\pi}{3}$

AMC 10 2011 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve
Strategy Introduce a Variable

The track length seems to depend on the unknown straight-side length and the unknown inner radius, so Tool #4 (Introduce a Variable) names them L and r and writes each edge as a formula. Tool #1 (Draw a Diagram) shows why the outer semicircles have radius r+6 while the straights stay the same. The magic is that both L and r cancel when we subtract, leaving a pure number. Tool #8 (Analyze the Units) then turns 'extra distance over speed equals extra time' into one clean equation for v.

1STEP 1

Name the parts and write the inner edge

Let v be the speed, L one straight side, r the inner end radius. Two inner semicircles make one circle, so the inner lap is 2L + 2π r.

P_inner = 2L + 2π r
2STEP 2

Write the outer edge

Draw the outer edge 6 meters out: the straights are still L each, but each end curve now has radius r + 6, so the outer lap is 2L + 2π(r+6).

P_outer = 2L + 2π (r + 6)
3STEP 3

Subtract to find the extra distance

Subtract the two laps: 2L and 2π r both cancel, leaving 2π · 6 = 12π meters of extra distance, whatever L and r happen to be.

P_outer - P_inner = 2π(r+6) - 2π r = 12π
4STEP 4

Turn the extra time into an equation

Both laps use the same v, so time is distance divided by v. The extra 12π meters is exactly the extra 36 seconds: 12π/v = 36.

12π/v = 36
5STEP 5

Solve for the speed

Solve 12π/v = 36: multiply by v, divide by 36, and v = 12π/36 = π/3 meters per second — choice (A).

v = 12π/36 = π/3 → (A)
Answer
π/3
Check the units and size: π/3 ≈ 1.05 meters per second is a normal walking pace. Multiplying back, the extra distance is v × 36 = π/3 × 36 = 12π meters, which matches the 12π we found from the geometry. The straight sides never mattered, which fits the fact that the answer choices carry no L.
💡Key takeaway

On a stadium track only the curved ends grow when you step outward, so the outer lap is always 2π × width longer — the straightaways don't matter.

  • Name the parts and write the inner edge
  • Write the outer edge
  • Subtract to find the extra distance
  • Turn the extra time into an equation
  • Solve for the speed

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