Answer :
To determine the value of [tex]\( x \)[/tex] for the given problem, we need to follow a systematic approach.
1. Understand Exterior Angles of a Regular Polygon:
The sum of the exterior angles of any polygon is always [tex]\( 360^\circ \)[/tex]. For a regular polygon (a polygon with all sides and angles equal), each exterior angle can be calculated by dividing [tex]\( 360^\circ \)[/tex] by the number of sides [tex]\( n \)[/tex].
2. Calculate the Exterior Angle for a Regular Decagon:
A regular decagon has 10 sides. Thus, each exterior angle of a regular decagon can be calculated as:
[tex]\[ \text{Exterior angle} = \frac{360^\circ}{10} = 36^\circ \][/tex]
3. Set Up the Equation:
We are given that each exterior angle is [tex]\((3x + 6)^\circ\)[/tex]. Therefore, we can set up the equation:
[tex]\[ 3x + 6 = 36 \][/tex]
4. Solve for [tex]\( x \)[/tex]:
To find [tex]\( x \)[/tex], we need to solve the equation [tex]\( 3x + 6 = 36 \)[/tex]:
[tex]\[ 3x + 6 = 36 \][/tex]
Subtract 6 from both sides:
[tex]\[ 3x = 30 \][/tex]
Divide both sides by 3:
[tex]\[ x = 10 \][/tex]
So, the value of [tex]\( x \)[/tex] is [tex]\( 10 \)[/tex].
Thus, the correct answer is:
[tex]\[ x = 10 \][/tex]
which matches the solution provided.
1. Understand Exterior Angles of a Regular Polygon:
The sum of the exterior angles of any polygon is always [tex]\( 360^\circ \)[/tex]. For a regular polygon (a polygon with all sides and angles equal), each exterior angle can be calculated by dividing [tex]\( 360^\circ \)[/tex] by the number of sides [tex]\( n \)[/tex].
2. Calculate the Exterior Angle for a Regular Decagon:
A regular decagon has 10 sides. Thus, each exterior angle of a regular decagon can be calculated as:
[tex]\[ \text{Exterior angle} = \frac{360^\circ}{10} = 36^\circ \][/tex]
3. Set Up the Equation:
We are given that each exterior angle is [tex]\((3x + 6)^\circ\)[/tex]. Therefore, we can set up the equation:
[tex]\[ 3x + 6 = 36 \][/tex]
4. Solve for [tex]\( x \)[/tex]:
To find [tex]\( x \)[/tex], we need to solve the equation [tex]\( 3x + 6 = 36 \)[/tex]:
[tex]\[ 3x + 6 = 36 \][/tex]
Subtract 6 from both sides:
[tex]\[ 3x = 30 \][/tex]
Divide both sides by 3:
[tex]\[ x = 10 \][/tex]
So, the value of [tex]\( x \)[/tex] is [tex]\( 10 \)[/tex].
Thus, the correct answer is:
[tex]\[ x = 10 \][/tex]
which matches the solution provided.
