AMC 8 · 2002 · #6

Grade 6 rate-ratio

Archetype Small-Domain Constrained Search └ Sub-archetype Factor-Pair Enumeration

rategraph-readingpattern-recognition pattern-recognitionidentify-subproblems ↑ Prerequisites: rate

📏 Short solution 💡 2 insights 📊 Diagram

Problem

A birdbath is filled by water flowing in at 20 ml/min while water drains out at 18 ml/min. Once the birdbath is full, extra water overflows. Of the five Volume-vs-Time graphs labeled A-E, pick the one that matches the birdbath from the moment it starts filling until well into the overflow stage.

Pick an answer.

(A)

(B)

(C)

(D)

(E)

AMC 8 2002 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Identify Subproblems

The story has two clearly separate phases, so Tool #7 (Identify Subproblems) splits the picture into Phase 1 (filling, before the bath is full) and Phase 2 (overflowing, after the bath is full). In each phase the net rate is constant, so each phase is a straight line; the slope just changes at the moment the bath fills up. Tool #1 (Draw a Diagram) then sketches the expected shape — a positive-slope ray from the origin followed by a horizontal segment — and we match that two-piece silhouette against the five options.

1STEP 1

Filling phase: inflow 20 ml/min minus drain 18 ml/min gives a net rise of 2 ml/min.

net rate = 20 - 18 = 2 ml/min

2STEP 2

A steady 2 ml/min rise adds equal volume each minute, so the graph is a straight line with positive slope from the origin.

V(t) = 2t for 0 ≤ t ≤ t_full

3STEP 3

Overflow phase: once full, the spare 2 ml/min spills over the edge, so the volume inside stays pinned at capacity.

V(t) = V_max for t ≥ t_full

4STEP 4

Stitch the phases: a rising ray, then a flat top. Only graph A shows a positive slope followed by a horizontal line.

shape = ↗ then → → (A)

Answer

Quick rule-out of the other choices using the two-phase shape. (B) starts flat (volume not yet rising) and then falls (volume cannot fall while inflow exceeds outflow) — wrong. (C) is a single rising line that never levels off, so the bath would never overflow — wrong. (D) is a flat line at a positive volume from the start, which says the bath is already full at t = 0 — wrong. (E) rises and then falls, suggesting the bath empties on its own — wrong, because the drain alone is slower than the inflow. Only (A) starts at the origin, climbs at a constant positive slope, and then flattens — matching both phases of the story.

💡Key takeaway

Split the story into two phases — filling (constant positive rate, so a straight line going up) and overflowing (volume stuck at capacity, so a horizontal line) — and only graph (A) shows that two-piece shape.

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