Question

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? 5.5π + 106 square inches 5.5π + 116 square inches 5.5π + 153 square inches 5.5π + 162 square inches 6 in. 48 in. 1 in.

Explanation:

Step1: Calculate the length of the two vertical parts

The length of each vertical part of the frame is 48 inches and there are 2 of them. Also, considering the 1 - inch width of the frame, the total length for the two vertical parts is $2\times48 = 96$ square - inches (since we are calculating the area of the wood, and for the vertical parts, it's length times width of the frame).

Step2: Calculate the area of the horizontal part at the bottom

The width of the frame is 1 inch. The length of the horizontal part at the bottom (across the width of the window) is $6 + 2\times1=8$ inches. So the area of the bottom horizontal part is $8\times1 = 8$ square - inches.

Step3: Calculate the area of the semi - circular part

The outer radius of the semi - circular part of the frame $R=\frac{6 + 2\times1}{2}=4$ inches and the inner radius $r = \frac{6}{2}=3$ inches. The area of a semi - circle is $A=\frac{1}{2}\pi(R^{2}-r^{2})$. So $A=\frac{1}{2}\pi(4^{2}-3^{2})=\frac{1}{2}\pi(16 - 9)=\frac{7}{2}\pi=3.5\pi$ square - inches. But we also have the straight part of the semi - circular frame which is 1 inch wide and its length is the diameter of the inner semi - circle which is 6 inches. So the area of this part is $6\times1 = 6$ square - inches. The total area for the semi - circular part of the frame is $3.5\pi+6 = 5.5\pi$ square - inches.

Step4: Calculate the total area of the wood

Add up the areas from the vertical, bottom - horizontal and semi - circular parts. The total area $A=96 + 8+5.5\pi=5.5\pi + 106$ square - inches.

Answer:

$5.5\pi + 106$ square inches