Competition · AMC preparation · step 4 of 4
AMC 8 · 2004 · #23
Grade 8 rate-ratio
Pick an answer.
AMC 8 2004 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
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.
Mark the distance at each corner
Sketch JKLM and mark the distance from J at each corner in order: 0, side JK, diagonal JL (the farthest), side JM, then 0 again.
Plotting distance-from-home at the four corners on coordinate axes is Grade 5 "graph real-world points" work.
5.G.A.2Draw A DiagramList three features of the graph
Turn those into three rules for the graph: start and end at 0, one single peak at the midpoint, and never flat anywhere.
Describing a function as "increases, then decreases, with one maximum" is the Grade 8 qualitative-graph reading move.
Tess's straight-line distance from home rises to a single high point partway through the run and then falls all the way back to zero, and along the way it never stays flat.
▸ Why?
She runs away from home and then all the way back, so her distance starts at zero, grows to its greatest value at the one point farthest from home, and shrinks back to zero — a single rise and fall with exactly one peak.
▸ Why?
Reading her distance from home as the radius of the circle around home that passes through where she is, the corner diagonally opposite home lies on a larger such circle than any other point of the block, so it is the single farthest point and the distance peaks there.
▸ Why?
Her distance from home is changing at every instant and never holds steady, because holding steady would mean moving along a curve whose every point is equally far from home, yet the block's sides are straight lines, not arcs bent around home.
Rule out A
Reject (A): it only climbs and never returns to 0, but Tess runs back home.
Reading the right-hand endpoint off a coordinate graph is Grade 8 function-interpretation work.
8.F.B.5Eliminate PossibilitiesRule out B
Reject (B): its flat stretches mean constant distance, which needs a circle around J — but she runs straight sides.
A horizontal segment means the function is constant on that interval — Grade 8 qualitative reading.
8.F.B.5Eliminate PossibilitiesRule out C
Reject (C): two peaks would mean the farthest point is hit twice, but only L is farthest.
Counting maxima on a graph is Grade 8 function-feature reading.
8.F.B.5Eliminate PossibilitiesRule out E
Reject (E): it has flat stretches and never returns to 0 — it breaks two rules at once.
Same Grade 8 check — flat means constant, and the right end must touch 0.
8.F.B.5Eliminate PossibilitiesConfirm D
Confirm (D): it rises to one midpoint peak, falls back to 0, splits into four segments, and never flattens — every rule holds.
After three failed tests, the last survivor is the answer — Grade 8 graph interpretation seals it.
8.F.B.5Eliminate PossibilitiesWhen 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!
- Mark the distance at each corner
- List three features of the graph
- Rule out A
- Rule out B
- Rule out C
- Rule out E
- Confirm D
A parent dashboard for the family lives at sensimlab.com.