AMC 8 · 2004 · #23

Grade 8 rate-ratio

Archetype Path Length Comparison & Shortest Path

graph-readingspatial-visualizationpattern-recognitionpythagorean-theorem pattern-recognitionpath-length-comparison ↑ Prerequisites: spatial-visualizationgraph-reading

📏 Medium solution 💡 3 insights 📊 Diagram

Problem

Tess runs counterclockwise around rectangular block $JKLM$ . She lives at corner $J$ . Which graph could represent her straight-line distance from home?

Pick an answer.

(A)

Distance increases steadily for the entire trip

(B)

Distance rises and falls in stairstep jumps with flat plateaus

(C)

Distance rises to a long flat plateau, then falls (rounded arcs)

(D)

Distance rises to a single peak in the middle, then falls back to 0

(E)

Distance rises, levels off, rises again, then levels off

View mode:

Toolkit + CCSS Solution

Understand

Restated: Tess runs counterclockwise around the rectangular block $JKLM$, starting and ending at her home corner $J$. We must pick the graph (out of A–E) whose vertical axis — Tess's straight-line distance from $J$ — rises, peaks at the corner opposite $J$, and falls back to $0$ as time goes by.

Givens: $JKLM$ is a rectangle with Tess's home at corner $J$; Tess runs at a steady speed counterclockwise: $J \to K \to L \to M \to J$; The vertical axis on each candidate graph is straight-line distance from $J$ (not distance run); Five candidate graphs A–E are sketched with different shapes

Unknowns: Which lettered graph (A, B, C, D, or E) correctly represents her distance from home over time

Understand

Plan

Primary tool: #3 Eliminate Possibilities

Secondary: #1 Draw a Diagram, #9 Solve an Easier Related Problem

Five concrete graphs are offered, so Tool #3 (Eliminate Possibilities) is the natural AMC multiple-choice move: list three qualitative features the right graph must have, then strike any graph that fails one. Tool #1 (Draw a Diagram) makes those features visible — sketching the rectangle and marking distances at the four corners reveals the rise-peak-fall shape. Tool #9 (Solve an Easier Related Problem) replaces the curved Pythagorean pieces with their easier endpoint values: we only need the corner distances ($0$, side $JK$, diagonal $JL$, side $JM$, $0$), not a calculus-style formula.

Execute — Answer: D

#1 Draw a Diagram 5.G.A.2 Step 1

Draw the rectangle $JKLM$ and mark Tess's straight-line distance from $J$ at each corner she reaches in order.
At $J$ the distance is $0$; at $K$ it is the side length $JK$; at $L$ it is the diagonal $JL$ (the farthest she ever gets); at $M$ it is the side length $JM$; back at $J$ it is $0$ again.

$$J \to 0,\; K \to JK,\; L \to JL,\; M \to JM,\; J \to 0$$

💡 Plotting distance-from-home at the four corners on coordinate axes is Grade 5 "graph real-world points" work.

#9 Solve an Easier Related Problem 8.F.B.5 Step 2

Use the corner distances to write down three must-have features for the correct graph.
(1) It starts at $0$ and ends at $0$.
(2) It has a single maximum value, the diagonal, reached in the middle of the trip — once Tess has gone halfway around (to $L$).
(3) It strictly rises from $J$ to $L$ and strictly falls from $L$ back to $J$ — no flat (constant-distance) stretch, because $J$ is a corner, not the center of a circle.

$$\text{shape: } 0 \nearrow JL \searrow 0,\; \text{single peak at } t = t_{\text{mid}}$$

💡 Describing a function as "increases, then decreases, with one maximum" is the Grade 8 qualitative-graph reading move.

#3 Eliminate Possibilities 8.F.B.5 Step 3

Eliminate (A).
Graph (A) is a single straight line that keeps rising all the way to the end.
That violates the must-end-at-$0$ rule — Tess goes back home, so her distance has to come back down.

$$\text{(A) ends above } 0 \;\Rightarrow\; \text{rejected}$$

💡 Reading the right-hand endpoint off a coordinate graph is Grade 8 function-interpretation work.

#3 Eliminate Possibilities 8.F.B.5 Step 4

Eliminate (B).
Graph (B) has flat horizontal pieces — long stretches of "distance does not change." Tess's distance from $J$ would only stay constant if she were running on a circle centered at $J$, but she runs on straight sides of a rectangle whose corner is $J$.
So her distance changes every moment.

$$\text{(B) has horizontal segments} \;\Rightarrow\; \text{rejected}$$

💡 A horizontal segment means the function is constant on that interval — Grade 8 qualitative reading.

#3 Eliminate Possibilities 8.F.B.5 Step 5

Eliminate (C).
Graph (C) shows two separate peaks with a flat region in between.
The path has only one farthest point ($L$), so there can be only one maximum.
Two peaks would mean Tess hits her greatest distance twice, which is impossible.

$$\text{(C) has two maxima} \;\Rightarrow\; \text{rejected}$$

💡 Counting maxima on a graph is Grade 8 function-feature reading.

#3 Eliminate Possibilities 8.F.B.5 Step 6

Eliminate (E).
Graph (E) also has flat horizontal pieces (constant distance), and it never comes back down to $0$.
Both features clash with the rules from Step 2.

$$\text{(E) has horizontals and ends above } 0 \;\Rightarrow\; \text{rejected}$$

💡 Same Grade 8 check — flat means constant, and the right end must touch $0$.

#3 Eliminate Possibilities 8.F.B.5 Step 7

Confirm (D).
Graph (D) starts at $0$, rises to a single peak at the middle of the time interval, then falls back to $0$.
It also shows four piecewise segments — one per side of the rectangle — and never flattens.
Every must-have feature checks out, so (D) is the only graph that could represent Tess's distance from home.

$$\text{(D) matches all three features} \;\Rightarrow\; \textbf{(D)}$$

💡 After three failed tests, the last survivor is the answer — Grade 8 graph interpretation seals it.

[1] #1 5.G.A.2 Draw the rectangle $JKLM$ and mark Tess's straight-line distance from $J$ at eac

[2] #9 8.F.B.5 Use the corner distances to write down three must-have features for the correct

[3] #3 8.F.B.5 Eliminate (A). Graph (A) is a single straight line that keeps rising all the way

[4] #3 8.F.B.5 Eliminate (B). Graph (B) has flat horizontal pieces — long stretches of "distanc

[5] #3 8.F.B.5 Eliminate (C). Graph (C) shows two separate peaks with a flat region in between.

[6] #3 8.F.B.5 Eliminate (E). Graph (E) also has flat horizontal pieces (constant distance), an

[7] #3 8.F.B.5 Confirm (D). Graph (D) starts at $0$, rises to a single peak at the middle of th

Review

Reasonableness: Sanity-check with a $6 \times 4$ block: side $JK = 4$, diagonal $JL = \sqrt{4^2+6^2} \approx 7.21$, side $JM = 6$. The corner distances $0, 4, 7.21, 6, 0$ rise to a single peak in the middle and fall back to $0$ — exactly the shape of graph (D). The two side-runs ($JK$ and $MJ$) give straight-line pieces on the graph, and the two cross-runs ($KL$ and $LM$) give pieces that curve gently toward the peak. None of them is ever flat, which is why graphs (B), (C), (E) had to go.

Alternative: Tool #10 (Make It Physical) is a fast classroom check: have someone walk counterclockwise around a rectangular table while a partner holds a measuring tape from the starting corner. The tape length grows to its longest at the opposite corner, then shrinks back to $0$ — a single rise-and-fall with no flat stretches. Among A–E only (D) shows that shape.

CCSS standards used (min grade 8)

5.G.A.2 Represent real-world and mathematical problems by graphing points in the first quadrant (Plotting Tess's distance from $J$ at each corner on a time–distance coordinate grid to set up the qualitative shape of the graph.)
8.F.B.5 Describe qualitatively the functional relationship between two quantities by analyzing a graph (Reading each candidate graph for increasing/decreasing intervals, constant (flat) segments, number of maxima, and start/end values, then eliminating any graph whose qualitative features clash with the rise-peak-fall story.)

⭐ When the answer choices are graphs, list two or three must-have features (starts at $0$, one peak, ends at $0$) and cross off any graph that breaks even one — Grade 8 qualitative graph reading is enough!

Problem

Toolkit + CCSS Solution

Understand

Plan

Execute — Answer: D

Review

CCSS standards used (min grade 8)

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