To determine the amount of wood needed for the carpenter to build the window frame, we need to evaluate the provided expressions and select the correct value based on the calculations.
Given expressions are:
1. [tex]\( 5.5 \pi + 106 \)[/tex]
2. [tex]\( 5.5 \pi + 116 \)[/tex]
3. [tex]\( 5.5 \pi + 153 \)[/tex]
4. [tex]\( 5.5 \pi + 162 \)[/tex]
Let's determine the numerical values of these expressions:
### Step-by-Step Evaluation:
1. For the expression [tex]\( 5.5 \pi + 106 \)[/tex]:
- Calculate [tex]\( 5.5 \pi \)[/tex]:
[tex]\[
5.5 \pi \approx 5.5 \times 3.141592653589793 \approx 17.27875959474386
\][/tex]
- Add 106 to the above result:
[tex]\[
17.27875959474386 + 106 \approx 123.27875959474386
\][/tex]
2. For the expression [tex]\( 5.5 \pi + 116 \)[/tex]:
- Using the same earlier calculated [tex]\( 5.5 \pi \)[/tex]:
[tex]\[
5.5 \pi \approx 17.27875959474386
\][/tex]
- Add 116 to the above result:
[tex]\[
17.27875959474386 + 116 \approx 133.27875959474386
\][/tex]
3. For the expression [tex]\( 5.5 \pi + 153 \)[/tex]:
- Using the same earlier calculated [tex]\( 5.5 \pi \)[/tex]:
[tex]\[
5.5 \pi \approx 17.27875959474386
\][/tex]
- Add 153 to the above result:
[tex]\[
17.27875959474386 + 153 \approx 170.27875959474386
\][/tex]
4. For the expression [tex]\( 5.5 \pi + 162 \)[/tex]:
- Using the same earlier calculated [tex]\( 5.5 \pi \)[/tex]:
[tex]\[
5.5 \pi \approx 17.27875959474386
\][/tex]
- Add 162 to the above result:
[tex]\[
17.27875959474386 + 162 \approx 179.27875959474386
\][/tex]
### Final Results:
- [tex]\( 5.5 \pi + 106 \approx 123.27875959474386 \)[/tex] square inches
- [tex]\( 5.5 \pi + 116 \approx 133.27875959474386 \)[/tex] square inches
- [tex]\( 5.5 \pi + 153 \approx 170.27875959474386 \)[/tex] square inches
- [tex]\( 5.5 \pi + 162 \approx 179.27875959474386 \)[/tex] square inches
Thus, the numerical results for each given expression are:
1. [tex]\( 123.27875959474386 \)[/tex]
2. [tex]\( 133.27875959474386 \)[/tex]
3. [tex]\( 170.27875959474386 \)[/tex]
4. [tex]\( 179.27875959474386 \)[/tex]
The amount of wood needed for each option is calculated and listed above. From these results, the carpenter can choose the appropriate amount of wood according to the specific requirements of the window frame design.
