To determine the value of [tex]\(w\)[/tex] given that one angle of a regular decagon measures [tex]\((8w + 17)^\circ\)[/tex], let's follow the step-by-step process:
1. Calculate the internal angle of a regular decagon:
- A decagon has 10 sides.
- The formula for the internal angle of a regular [tex]\(n\)[/tex]-sided polygon is [tex]\(\frac{(n - 2) \times 180}{n}\)[/tex].
2. Apply the formula to the decagon:
[tex]\[
\text{Internal angle} = \frac{(10 - 2) \times 180}{10} = \frac{8 \times 180}{10} = \frac{1440}{10} = 144^\circ
\][/tex]
3. Set up the equation with the given internal angle:
- The internal angle [tex]\(144^\circ\)[/tex] is given by [tex]\((8w + 17)^\circ\)[/tex].
- We set up the equation:
[tex]\[
8w + 17 = 144
\][/tex]
4. Solve for [tex]\(w\)[/tex]:
[tex]\[
8w + 17 = 144
\][/tex]
- Subtract 17 from both sides:
[tex]\[
8w = 144 - 17
\][/tex]
[tex]\[
8w = 127
\][/tex]
- Divide both sides by 8 to solve for [tex]\(w\)[/tex]:
[tex]\[
w = \frac{127}{8} \approx 15.875
\][/tex]
5. Round to the nearest whole number:
[tex]\[
15.875 \approx 16
\][/tex]
Therefore, the value of [tex]\(w\)[/tex] that makes one angle of a regular decagon measure [tex]\((8w + 17)^\circ\)[/tex] is [tex]\( \boxed{16} \)[/tex].
