Problem 7 · 2023 AMC 8
Medium
Geometry & Measurement
evaluate-formulagrid
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Answer: B — 1 point.
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Hint 1 of 2
The rectangle is tiny and way off to the right (x only runs 15 to 16). So instead of graphing the whole lines, just ask: when a line reaches that far right, is its height between 3 and 5? You only need to test x = 15 and x = 16.
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Hint 2 of 2
Read each line's slope off its two points to get its equation. Line AB rises 1 for every 3 right: y = x/3. Line CD drops 1 for every 2 right: y = 10 − x/2. Plug in x = 15 and 16.
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Approach: evaluate the two lines at the rectangle's x-range
- Notice the rectangle is a thin sliver far to the right: x runs only 15 to 16, y only 3 to 5. So you don't graph the whole lines — you just check whether either line is at the right height when it gets out to x = 15 or 16.
- Line AB: slope = 1/3 (from A(0,0) to B(3,1)), so y = x/3. At x = 15: y = 5 — exactly the corner (15, 5) ✓. At x = 16: y ≈ 5.33, just above the box.
- Line CD: slope = −1/2, so y = 10 − x/2. At x = 15: y = 2.5; at x = 16: y = 2 — both below the box.
- Only (15, 5) lands on the rectangle — 1 point. Why this is faster: when a region is far from the lines, test the region's edges rather than tracing the lines all the way out.
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