Asked by bobo
carpenter is making a wooden window frame that has a width of 1 inch.
A window with a one-inch frame is shown. The frame is comprised of a rectangle and a semicircle. The rectangle has side lengths of 12 inches and 48 inches. The semicircle has a radius of 6 inches. The frame is 1-inch wider than the window.
How much wood does the carpenter need to build the frame?
5.5 + 106 square inches
5.5 + 116 square inches
5.5 + 153 square inches
5.5 + 162 square inches
All Answers 1
Answered by
GPT-5 mini
AI
Compute the frame area as outer area minus inner area. For a planar shape dilated outward by r the area increase is P·r + πr^2 (Steiner’s formula). Here r = 1, so wood area = P + π.
Find the original perimeter P. The window is a 12×48 rectangle with a semicircle (radius 6) attached to the 12‑inch side, so
P = bottom 12 + two verticals 48+48 + semicircular arc (½ circumference) = 12 + 96 + 6π = 108 + 6π.
Thus wood area = P + π = (108 + 6π) + π = 108 + 7π square inches ≈ 129.99 in^2.
None of the four listed choices equals 108 + 7π, so the correct amount of wood is 108 + 7π (≈130) square inches.
Find the original perimeter P. The window is a 12×48 rectangle with a semicircle (radius 6) attached to the 12‑inch side, so
P = bottom 12 + two verticals 48+48 + semicircular arc (½ circumference) = 12 + 96 + 6π = 108 + 6π.
Thus wood area = P + π = (108 + 6π) + π = 108 + 7π square inches ≈ 129.99 in^2.
None of the four listed choices equals 108 + 7π, so the correct amount of wood is 108 + 7π (≈130) square inches.
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