Asked by julie
A 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
Let the inner window be the rectangle 12 by 48 with a semicircle of radius 6 on one 12-inch side.
Perimeter of the inner window (the boundary that the frame goes around) is
P = bottom 12 + two vertical sides 2·48 + semicircular arc (½ circumference) = 12 + 96 + 6π = 108 + 6π.
Dilating the shape by 1 inch (frame thickness 1) increases area by P·1 + π·1^2 (Steiner formula for offsetting by a disk of radius 1). Thus the wood area is
Area(frame) = P + π = (108 + 6π) + π = 108 + 7π square inches ≈ 129.99 in^2.
None of the four listed choices equals 108 + 7π.
Perimeter of the inner window (the boundary that the frame goes around) is
P = bottom 12 + two vertical sides 2·48 + semicircular arc (½ circumference) = 12 + 96 + 6π = 108 + 6π.
Dilating the shape by 1 inch (frame thickness 1) increases area by P·1 + π·1^2 (Steiner formula for offsetting by a disk of radius 1). Thus the wood area is
Area(frame) = P + π = (108 + 6π) + π = 108 + 7π square inches ≈ 129.99 in^2.
None of the four listed choices equals 108 + 7π.
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