Four top rectangles match five bottom ones in length, giving 4l = 5b, so l = 5b/4. Area 9lb = 72 yields lb = 8. Solving gives b = 2.53 cm, l = 3.16 cm, and perimeter ≈ 11.38 cm.
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No, a larger stagger is needed. In a 400 m race, runners navigate two bends on a 400 m track but four bends on a 200 m track, requiring twice the total stagger distance.
Let OA = OB = r. Semicircle on diameter AB has radius r/√2 and area (1/4)πr². Subtracting circular segment AFB from both the semicircle and the quarter circle OAB leaves equal areas for both shaded regions.
By Pythagoras theorem, leg semicircles sum to the hypotenuse semicircle: Semicircle(a) + Semicircle(b) = Semicircle(c). Subtracting the two unshaded circular segments from both sides shows that the lunes’ area Area(A) + Area(B) equals the triangle’s area Area(C).
The fraction of the rectangle covered by n identical tangent circles in a row is always constant at π / 4 approximately 11/14 (or 78.5%) regardless of whether n equals 10, 20 or 50.