AMC 10 · 2011 · #8

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
Walking the outer edge of a track of known width takes a fixed extra time. Find the walking speed.

Pick an answer.

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

Both edges share the same straight parts.

P_inner = 2L + 2π r
2STEP 2

Write the outer edge

The outer edge only differs at the curves.

P_outer = 2L + 2π (r + 6)
3STEP 3

Subtract to find the extra distance

Subtracting leaves a distance free of the radius.

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

Turn the extra time into an equation

The extra time becomes one equation.

12π/v = 36
5STEP 5

Solve for the speed

Solving gives π/3, 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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