To determine the missing quantity [tex]\( r \)[/tex] in the problem, let’s go through the steps required to find it.
We are given:
- The central angle [tex]\( \theta = \frac{1}{2} \)[/tex] radians,
- The arc length [tex]\( s = 5 \)[/tex] feet,
and we need to find the radius [tex]\( r \)[/tex] of the circle.
The formula that relates the arc length [tex]\( s \)[/tex], the radius [tex]\( r \)[/tex], and the central angle [tex]\( \theta \)[/tex] is:
[tex]\[ s = r \theta \][/tex]
We need to solve for [tex]\( r \)[/tex]. Rearrange the formula to isolate [tex]\( r \)[/tex]:
[tex]\[ r = \frac{s}{\theta} \][/tex]
Now, substitute the given values [tex]\( s = 5 \)[/tex] feet and [tex]\( \theta = \frac{1}{2} \)[/tex] radians into the equation:
[tex]\[ r = \frac{5 \text{ feet}}{\frac{1}{2} \text{ radian}} \][/tex]
To simplify the division, remember that dividing by a fraction is the same as multiplying by its reciprocal:
[tex]\[ r = 5 \text{ feet} \times \frac{2}{1} \][/tex]
This simplifies to:
[tex]\[ r = 10 \text{ feet} \][/tex]
So, the radius [tex]\( r \)[/tex] of the circle is [tex]\( 10 \)[/tex] feet.
