Competition · AMC preparation · step 4 of 4

AMC 8 · 2023 · #9

Grade 5 rate-ratio
Archetype Coordinate Plane Figure Computation └ Sub-archetype Coordinate Graph Slope as Rate
graph-readingcoordinate-geometryinterval-arithmetic graph-readingidentify-subproblems ↑ Prerequisites: coordinate-geometrymulti-digit-arithmetic
📏 Medium solution 💡 2 insights 📊 Diagram
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Problem
A graph shows Malaika's elevation (in meters) above the base as time (in seconds) passes while she skis. We need the total number of seconds her elevation E stays inside the band 4 ≤ E ≤ 7 meters.

Pick an answer.

(A)
6
(B)
8
(C)
10
(D)
12
(E)
14

AMC 8 2023 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve
Strategy Draw a Diagram

The problem already gives a graph, so the heavy lifting is reading it carefully — Tool #1 (Draw/use a diagram) applies directly: we add two horizontal reference lines y=4 and y=7 on top of the given curve and read off where they cross. Once those crossings split the timeline into pieces, Tool #7 (Subproblems) lets us treat each piece independently — duration of piece 1, plus duration of piece 2, and so on — instead of trying to handle the whole curve at once. Finally, since this is multiple-choice, Tool #3 confirms the total against the five listed options.

1STEP 1

Mark the band on the graph

Draw two horizontal reference lines at y = 4 and y = 7; the strip between them is the target band whose time we must total.

band: 4 ≤ y ≤ 7
2STEP 2

Read off the crossing times

Read the crossings: the curve meets y = 7 at t = 2, 6, 10, 14 and meets y = 4 at t = 4 and 12 seconds.

y=7colon t = 2, 6, 10, 14; y=4colon t = 4, 12
3STEP 3

Split the timeline into intervals

Four in-band pieces appear: [2,4], [4,6], [10,12], [12,14]; the t=6 to t=10 peak sits above 7 m, so it is excluded.

in-band intervals: [2,4], [4,6], [10,12], [12,14]
4STEP 4

Add the interval lengths

Each piece spans 2 seconds and there are four of them: 2 + 2 + 2 + 2 = 8 seconds inside the band.

(4-2) + (6-4) + (12-10) + (14-12) = 2+2+2+2 = 8
5STEP 5

Match to the choices

The total 8 seconds is choice (B); larger options would need in-band time between t=6 and t=10, where she is above 7 m.

8 seconds → (B)
Answer
8
Does 8 seconds make sense? The curve enters the [4,7] band four separate times and each crossing-to-crossing trip takes the same 2 seconds, so 4 × 2 = 8. The middle hump (where she is highest, above 7 m, from t=6 to t=10) and the deepest dip (where she is below 4 m, between t=4 and t=12 on the y=4 side except the band-band pieces) are correctly EXCLUDED. The answer (B) 8 is consistent.
💡Key takeaway

This AMC 8 problem only needs Grade 5 reading points on a coordinate graph you already know!

  • Mark the band on the graph
  • Read off the crossing times
  • Split the timeline into intervals
  • Add the interval lengths
  • Match to the choices

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