AMC 8 · 2002 · #6
Grade 6 rate-ratioProblem
A birdbath is designed to overflow so that it will be self-cleaning. Water flows in at the rate of 20 milliliters per minute and drains at the rate of 18 milliliters per minute. Which one of these graphs shows the volume of water in the birdbath during the filling time and continuing into the overflow time?
Pick an answer.
Toolkit + CCSS Solution
Understand
Restated: 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.
Givens: Inflow rate: $20$ ml/min (constant); Outflow rate (drain): $18$ ml/min (constant); The birdbath starts the story empty and fills, then overflows when it reaches capacity; Axes on every graph: horizontal $=$ time, vertical $=$ volume; Answer choices: graphs (A), (B), (C), (D), (E)
Unknowns: Which lettered graph shows volume vs. time for this birdbath
Understand
Restated: 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.
Givens: Inflow rate: $20$ ml/min (constant); Outflow rate (drain): $18$ ml/min (constant); The birdbath starts the story empty and fills, then overflows when it reaches capacity; Axes on every graph: horizontal $=$ time, vertical $=$ volume; Answer choices: graphs (A), (B), (C), (D), (E)
Plan
Primary tool: #7 Identify Subproblems
Secondary: #1 Draw a Diagram
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.
Execute — Answer: A
6.RP.A.3 Step 1 - Phase 1 (filling): compute the net rate of change of volume.
- Water comes in at $20$ ml/min and leaves at $18$ ml/min, so the volume grows at the difference.
💡 Combining two rates by subtraction is the Grade 6 "unit rate" move: the net effect is a single constant rate.
6.EE.C.9 Step 2 - Translate that constant rate into a graph shape.
- A constant positive rate of $2$ ml/min means the volume rises by the same amount each minute, so the Volume-vs-Time graph is a straight line with positive slope.
- The bath starts empty, so the line starts at the origin.
💡 Grade 6 "two-variable equations": a constant rate $\Rightarrow$ a linear relationship $\Rightarrow$ a straight-line graph through $(0,0)$.
6.RP.A.3 Step 3 - Phase 2 (overflowing): once the bath reaches capacity, any extra inflow has nowhere to go and simply spills over the edge.
- The drain still removes $18$ ml/min, and the overflow removes the remaining $2$ ml/min, so the volume inside the bath stays pinned at capacity.
💡 When inflow and total outflow balance, the volume stops changing — on the graph, that is a flat horizontal segment.
6.EE.C.9 Step 4 - Stitch the two phases together.
- The full graph is a ray from the origin with positive slope, then a horizontal segment at height $V_{\max}$.
- Match this two-piece silhouette to the choices: only graph A shows a positive-slope line followed by a horizontal line.
💡 Choosing a graph is a Grade 6 "match the story to the picture" task once the algebra of each phase is in hand.
6.RP.A.3 Phase 1 (filling): compute the net rate of change of volume. Water comes in at $ 6.EE.C.9 Translate that constant rate into a graph shape. A constant positive rate of $2$ 6.RP.A.3 Phase 2 (overflowing): once the bath reaches capacity, any extra inflow has nowh 6.EE.C.9 Stitch the two phases together. The full graph is a ray from the origin with pos Review
Reasonableness: 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.
Alternative: Tool #15 (Visualize / Reorganize): instead of writing equations, sketch the situation with three labels on the time axis — "empty", "just full", "long after full" — and ask what the volume must be at each. At "empty" it is $0$; at "just full" it is $V_{\max}$; at "long after full" it is still $V_{\max}$. Three points alone — $(0,0)$, $(t_{\text{full}}, V_{\max})$, $(2t_{\text{full}}, V_{\max})$ — already pick out (A) without ever computing the $2$ ml/min net rate.
CCSS standards used (min grade 6)
6.RP.A.3Use ratio and rate reasoning to solve real-world and mathematical problems (Combining the $20$ ml/min inflow and $18$ ml/min outflow into a single net rate of $2$ ml/min during filling, and recognizing that inflow and total outflow balance during overflow.)6.EE.C.9Use variables to represent two quantities that change in relationship to one another; write an equation; analyze the relationship (Writing $V = 2t$ for the filling phase and $V = V_{\max}$ for the overflow phase, and reading off the corresponding straight-line and horizontal-line graph segments.)
⭐ 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.
⭐ 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.