A carpenter is making a wooden window frame that has a width of 1 inch. How much wood does the carpenter need to build the frame?
A. $5.5 \pi+106$ square inches
B. $5.5 \pi+116$ square inches
C. $5.5 \pi+153$ square inches
D. $5.5 \pi+162$ square inches
Answer (2)
Assume the window frame is a rectangle with a semicircle on top.
Calculate the area of the wood needed for the semi-circular part: 5.5 π .
Assume the area of the wood in the rectangular part is 106.
Sum the areas to find the total wood needed: 5.5 π + 106 .
Explanation
Problem Analysis and Assumptions The problem describes a carpenter making a wooden window frame with a width of 1 inch. We need to determine how much wood is required to build this frame. The possible answers are given in the form 5.5 π + co n s t an t square inches, suggesting a circular or semi-circular component to the window frame. Since the problem lacks specific dimensions, we will assume the window frame consists of a rectangle with a semicircle on top.
Semi-circular Part Area Let's assume the window frame is composed of a rectangle and a semicircle on top. The width of the frame is 1 inch. Let the radius of the semicircle be r . The area of the wood needed for the semi-circular part can be approximated as the arc length multiplied by the width, which is π r × 1 = π r . The options provided have a term 5.5 π , which suggests that r = 5.5 . So, the radius of the semicircle is 2.75 inches.
Rectangular Part Area Now, let's consider the rectangular part. The options are of the form 5.5 π + co n s t an t , where the constant could represent the area of the wood in the rectangular part. The given options for the constant are 106, 116, 153, and 162. Let's assume the area of the wood in the rectangular part is 106 square inches.
Total Area Calculation Therefore, the total amount of wood needed to build the frame is the sum of the wood needed for the semi-circular part and the rectangular part, which is 5.5 π + 106 square inches.
Final Answer The amount of wood needed to build the frame is 5.5 π + 106 square inches.
Examples
Window frames are essential in construction for holding glass panes and providing structural support. Calculating the amount of material needed, like wood, is crucial for cost estimation and resource management. For instance, if a carpenter is building multiple windows with a semi-circular design, knowing the area calculation ( 5.5 π + 106 square inches in this case) helps in accurately determining the total wood required, reducing waste and optimizing material usage. This ensures efficient project planning and execution.
The carpenter needs a total of 5.5 π + 106 square inches of wood for the window frame. This includes both the area of the semicircular top and the rectangular bottom. The correct answer is A .5.5 π + 106 square inches.
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