To determine the amount of wood a carpenter needs to build a window frame, we are given four potential options for the area of the frame. We will analyze each option to find which one is correct based on our calculations.
The possible areas provided are as follows:
1. [tex]\(5.5 \pi + 106\)[/tex] square inches
2. [tex]\(5.5 \pi + 116\)[/tex] square inches
3. [tex]\(5.5 \pi + 153\)[/tex] square inches
4. [tex]\(5.5 \pi + 162\)[/tex] square inches
We start by calculating the value for each of these options using an appropriate value for [tex]\(\pi\)[/tex]. We'll use [tex]\(\pi \approx 3.14159\)[/tex] for our calculations.
### Calculations:
#### Option 1: [tex]\(5.5 \pi + 106\)[/tex]
[tex]\[
5.5 \pi + 106 \approx 5.5 \times 3.14159 + 106 = 17.278755 + 106 = 123.27875959474386 \text{ square inches}
\][/tex]
#### Option 2: [tex]\(5.5 \pi + 116\)[/tex]
[tex]\[
5.5 \pi + 116 \approx 5.5 \times 3.14159 + 116 = 17.278755 + 116 = 133.27875959474386 \text{ square inches}
\][/tex]
#### Option 3: [tex]\(5.5 \pi + 153\)[/tex]
[tex]\[
5.5 \pi + 153 \approx 5.5 \times 3.14159 + 153 = 17.278755 + 153 = 170.27875959474386 \text{ square inches}
\][/tex]
#### Option 4: [tex]\(5.5 \pi + 162\)[/tex]
[tex]\[
5.5 \pi + 162 \approx 5.5 \times 3.14159 + 162 = 17.278755 + 162 = 179.27875959474386 \text{ square inches}
\][/tex]
### Conclusion:
After performing these calculations, we find the respective areas as follows:
1. [tex]\(123.27875959474386\)[/tex] square inches
2. [tex]\(133.27875959474386\)[/tex] square inches
3. [tex]\(170.27875959474386\)[/tex] square inches
4. [tex]\(179.27875959474386\)[/tex] square inches
Given these, the carpenter needs a total area of:
[tex]\[
179.27875959474386 \text{ square inches}.
\][/tex]
Thus, the correct choice is:
[tex]\[
5.5 \pi + 162 \text{ square inches}.
\][/tex]
