To determine how much wood the carpenter needs to build the window frame, let's dissect the problem step-by-step.
1. Understanding the Circular Part:
The formula includes a term "5.5π". This term suggests that there is a circular component in the frame whose area we need to calculate.
2. Calculation of the Circular Component:
The term "5.5π" represents the area contributed by the circular part. Using π = 3.141592653589793, we can substitute and calculate this area:
[tex]\[
\text{Circular part area} = 5.5 \times \pi = 5.5 \times 3.141592653589793 = 17.27875959474386 \text{ square inches}
\][/tex]
3. Calculating Total Areas:
Next, we need to find the total area of the wooden window frame necessary for each of the provided expressions:
- For [tex]\(5.5\pi + 106\)[/tex]:
[tex]\[
\text{Total area} = 17.27875959474386 + 106 = 123.27875959474386 \text{ square inches}
\][/tex]
- For [tex]\(5.5\pi + 116\)[/tex]:
[tex]\[
\text{Total area} = 17.27875959474386 + 116 = 133.27875959474386 \text{ square inches}
\][/tex]
- For [tex]\(5.5\pi + 153\)[/tex]:
[tex]\[
\text{Total area} = 17.27875959474386 + 153 = 170.27875959474386 \text{ square inches}
\][/tex]
- For [tex]\(5.5\pi + 162\)[/tex]:
[tex]\[
\text{Total area} = 17.27875959474386 + 162 = 179.27875959474386 \text{ square inches}
\][/tex]
Thus, the carpenter needs the following amounts of wood, depending on the various conditions:
- For [tex]\(5.5\pi + 106\)[/tex]: [tex]\(123.27875959474386\)[/tex] square inches
- For [tex]\(5.5\pi + 116\)[/tex]: [tex]\(133.27875959474386\)[/tex] square inches
- For [tex]\(5.5\pi + 153\)[/tex]: [tex]\(170.27875959474386\)[/tex] square inches
- For [tex]\(5.5\pi + 162\)[/tex]: [tex]\(179.27875959474386\)[/tex] square inches
