To find the radius of a circle given its circumference, we use the formula for the circumference of a circle, which is:
[tex]\[ C = 2\pi r \][/tex]
where
- [tex]\( C \)[/tex] is the circumference,
- [tex]\( r \)[/tex] is the radius,
- [tex]\( \pi \)[/tex] (pi) is approximately 3.14159.
Given the circumference [tex]\( C = 5 \)[/tex] miles, we need to solve for the radius [tex]\( r \)[/tex].
First, write the formula again with the given value:
[tex]\[ 5 = 2\pi r \][/tex]
Next, solve for [tex]\( r \)[/tex] by isolating [tex]\( r \)[/tex] on one side of the equation. To do this, divide both sides of the equation by [tex]\( 2\pi \)[/tex]:
[tex]\[ r = \frac{5}{2\pi} \][/tex]
This fraction is the exact answer and represents the radius in its simplest form. So, the radius of the circle is:
[tex]\[ r = \frac{5}{2\pi} \][/tex] miles.
To find the approximate numerical value, we substitute [tex]\( \pi \approx 3.14159 \)[/tex] into the equation:
[tex]\[ r \approx \frac{5}{2 \times 3.14159} \][/tex]
[tex]\[ r \approx \frac{5}{6.28318} \][/tex]
[tex]\[ r \approx 0.79577 \][/tex] miles.
Thus, the exact radius of the circle is [tex]\( \frac{5}{2\pi} \)[/tex] miles, and its approximate value is 0.79577 miles.
