Problem 21 · 2012 AMC 8
Medium
Geometry & Measurement
surface-area-split
Marla has a large white cube that has an edge of 10 feet. She also has enough green paint to cover 300 square feet. Marla uses all the paint to create a white square centered on each face, surrounded by a green border. What is the area of one of the white squares, in square feet?
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Answer: D — 50 square feet.
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Hint 1 of 2
You never need the size or shape of the green border — just areas. Every bit of the cube's surface is either green paint or white square, so white = total surface − green.
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Hint 2 of 2
This is whole minus the part you know. Find the cube's total surface, subtract the 300 of green, then split the remaining white equally among the 6 identical faces. (The √2 answer choices are bait for finding a side length you don't need.)
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Approach: total surface area, subtract green, divide by 6
- A cube has 6 faces, each 10 × 10, so total surface = 6 × 100 = 600 sq ft.
- Green covers 300 of it, so the white squares together cover 600 − 300 = 300 sq ft.
- The 6 faces are identical, so each white square is 300 / 6 = 50 sq ft.
- Note: the question asks for the white area, so stop here — no need to take a square root to find the side. Watching what's actually asked saves the √2 traps.
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