A decagon has 10 sides. One angle of a regular decagon measures [tex][tex]$(8w + 17)^\circ$[/tex][/tex]. Determine the value of [tex]w[/tex]. Round to the nearest whole number.

A. [tex]w = 144[/tex]
B. [tex]w = 23[/tex]
C. [tex]w = 16[/tex]
D. [tex]w = 8[/tex]



Answer :

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].