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
Answer: 108 + 7π square inches (≈ 130.0 in²).
Work:
- The window boundary is the three straight sides of the 12×48 rectangle plus the semicircular arc. So its perimeter is 48 + 48 + 12 + (1/2)(2π·6) = 108 + 6π.
- For a convex planar shape, an outward offset by t adds area = perimeter·t + πt². With t = 1 this gives added area = (108 + 6π)·1 + π·1² = 108 + 7π.
Numerically 108 + 7π ≈ 108 + 21.99 ≈ 130.0 in². None of the four multiple-choice forms given (5.5 + 106, 5.5 + 116, 5.5 + 153, 5.5 + 162) equals 108 + 7π.
Work:
- The window boundary is the three straight sides of the 12×48 rectangle plus the semicircular arc. So its perimeter is 48 + 48 + 12 + (1/2)(2π·6) = 108 + 6π.
- For a convex planar shape, an outward offset by t adds area = perimeter·t + πt². With t = 1 this gives added area = (108 + 6π)·1 + π·1² = 108 + 7π.
Numerically 108 + 7π ≈ 108 + 21.99 ≈ 130.0 in². None of the four multiple-choice forms given (5.5 + 106, 5.5 + 116, 5.5 + 153, 5.5 + 162) equals 108 + 7π.
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