Problem 8 · 2003 AMC 8
Medium
Geometry & Measurement
area
Art, Roger, Paul, and Trisha bake cookies that are all the same thickness, in the shapes shown below (dimensions in inches). Each friend uses the same amount of dough, and Art's batch makes exactly 12 cookies.
Who makes the fewest cookies from one batch of dough?
Show answer
Answer: A — Art.
Show hints
Hint 1 of 2
Flip the question: the SAME pile of dough splits into fewer cookies exactly when each cookie is bigger — so you're really hunting for the largest single shape.
Still stuck? Show hint 2 →
Hint 2 of 2
Just compare the four areas; the biggest area means the fewest cookies.
Show solution
Approach: same dough, so biggest cookie means fewest cookies
- Everyone starts with the same amount of dough, so cookies = dough ÷ (one cookie's size). Bigger cookie, fewer of them. So you only need to find the largest shape — no need to know how much dough there is.
- Areas (square inches): Art is a trapezoid, ½(3 + 5)(3) = 12; Roger's rectangle 2 × 4 = 8; Paul's parallelogram 3 × 2 = 6; Trisha's triangle ½(3)(4) = 6.
- Art's 12 in² cookie is the biggest, so Art makes the fewest.
- Worth keeping: a trapezoid's area is ½(top + bottom) × height — the average of the two parallel sides times the distance between them.
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