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)
To calculate the amount of wood needed for a window frame shaped like a rectangle with a semicircle on top, we use the formula for area, involving width and height. By assuming a width of 11 inches, we find possible heights for given area options. Based on the calculations, one reasonable choice is 5.5 π + 116 square inches.
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Assume the window frame is a rectangle with a semicircle on top.
Express the area of the wood needed as A = w + 2 h + 2 π w , where w is the width and h is the height of the rectangular part.
Given the possible areas are in the form 5.5 π + co n s t an t , assume w = 11 .
Calculate the possible heights h for each given area. The possible areas are 5.5 π + 106 , 5.5 π + 116 , 5.5 π + 153 , and 5.5 π + 162 square inches.
Explanation
Problem Analysis and Assumptions The problem states that a carpenter is making a wooden window frame with a width of 1 inch. We need to determine how much wood is needed to build the frame, given four possible answers. Since the shape and dimensions of the window frame are not provided, we must make an assumption about its shape. Let's assume the window frame is a rectangle with a semicircle on top.
Setting up the Area Equation Let w be the width of the rectangular part of the window frame, and let h be the height of the rectangular part. The radius of the semicircle on top is then r = w /2 . The perimeter of the frame is given by the sum of the lengths of all sides, which is w + 2 h + π r = w + 2 h + π ( w /2 ) . Since the width of the frame is 1 inch, the area of the wood needed is equal to the perimeter. Therefore, the area A is given by: A = w + 2 h + 2 π w We are given four possible answers for the area: 5.5 π + 106 , 5.5 π + 116 , 5.5 π + 153 , and 5.5 π + 162 .
Solving for h in each case Notice that each answer is in the form 5.5 π + co n s t an t . This suggests that w /2 = 5.5 , so w = 11 . Substituting w = 11 into the area equation, we get: A = 11 + 2 h + 2 π ( 11 ) = 11 + 2 h + 5.5 π Now we can solve for h in each of the possible answers:
If A = 5.5 π + 106 , then 5.5 π + 106 = 11 + 2 h + 5.5 π , so 2 h = 106 − 11 = 95 , and h = 47.5 .
If A = 5.5 π + 116 , then 5.5 π + 116 = 11 + 2 h + 5.5 π , so 2 h = 116 − 11 = 105 , and h = 52.5 .
If A = 5.5 π + 153 , then 5.5 π + 153 = 11 + 2 h + 5.5 π , so 2 h = 153 − 11 = 142 , and h = 71 .
If A = 5.5 π + 162 , then 5.5 π + 162 = 11 + 2 h + 5.5 π , so 2 h = 162 − 11 = 151 , and h = 75.5 .
Final Answer Based on Assumptions Since we are asked to find the amount of wood needed, and we have assumed the window frame is a rectangle with a semicircle on top, with a width of 1 inch for the frame itself, the amount of wood needed is the perimeter of the outer frame. We found that if the width of the rectangular part is 11 inches, then the possible heights are 47.5, 52.5, 71, and 75.5 inches. The corresponding areas are 5.5 π + 106 , 5.5 π + 116 , 5.5 π + 153 , and 5.5 π + 162 square inches. Without more information, we cannot determine the exact amount of wood needed. However, since we made the assumption that the width of the rectangular part is 11, we can express the area as 5.5 π + co n s t an t , where the constant is 11 + 2 h . The possible values for the area are given in the problem.
Examples
When building a window frame, carpenters need to calculate the amount of wood required to construct the frame. This involves determining the perimeter of the window and multiplying it by the width of the wood used for the frame. For example, if a window frame is shaped like a rectangle with a semicircle on top, the carpenter would calculate the length of the rectangular sides plus the length of the semicircular arc to find the total perimeter. Multiplying this perimeter by the width of the wood gives the total amount of wood needed for the frame, ensuring they purchase the correct amount of materials for the job.
