To determine the radius of a circle given that a central angle of [tex]\(\frac{2\pi}{3}\)[/tex] radians intercepts an arc of [tex]\(35\pi\)[/tex] kilometers, we'll use the formula for the length of an arc in a circle. The formula is:
[tex]\[ L = r \theta \][/tex]
where:
- [tex]\( L \)[/tex] is the arc length,
- [tex]\( r \)[/tex] is the radius of the circle,
- [tex]\( \theta \)[/tex] is the central angle in radians.
Given the information:
- [tex]\( L = 35\pi \)[/tex] kilometers,
- [tex]\( \theta = \frac{2\pi}{3} \)[/tex] radians.
We need to solve for [tex]\( r \)[/tex]. Rearrange the formula to solve for the radius:
[tex]\[ r = \frac{L}{\theta} \][/tex]
Substituting the given values:
[tex]\[ r = \frac{35\pi}{\frac{2\pi}{3}} \][/tex]
To simplify the division of fractions, we multiply by the reciprocal:
[tex]\[ r = 35\pi \times \frac{3}{2\pi} \][/tex]
Notice that [tex]\(\pi\)[/tex] in the numerator and denominator cancels out:
[tex]\[ r = 35 \times \frac{3}{2} \][/tex]
[tex]\[ r = \frac{105}{2} \][/tex]
[tex]\[ r = 52.5 \][/tex]
Thus, the radius of the circle is:
[tex]\[ r = 52.5 \, \text{kilometers} \][/tex]
The correct answer is:
D. 52.5 km
