To find the radius of a circle given its circumference and the value of π, we can follow these steps:
1. Recall the formula for the circumference of a circle:
[tex]\[
C = 2 \pi r
\][/tex]
where [tex]\( C \)[/tex] is the circumference, [tex]\( \pi \)[/tex] (pi) is a constant approximately equal to 3.14159 (often the fraction [tex]\( \frac{22}{7} \)[/tex] is used for simplicity), and [tex]\( r \)[/tex] is the radius of the circle.
2. Insert the given values into the formula:
The circumference [tex]\( C \)[/tex] is given as 66 cm, and [tex]\( \pi \)[/tex] is given as [tex]\( \frac{22}{7} \)[/tex]. Our formula looks like this:
[tex]\[
66 = 2 \cdot \frac{22}{7} \cdot r
\][/tex]
3. Solve for the radius [tex]\( r \)[/tex]:
To isolate [tex]\( r \)[/tex], we need to manipulate the equation:
[tex]\[
66 = \frac{44}{7} \cdot r
\][/tex]
Multiply both sides by [tex]\( \frac{7}{44} \)[/tex] to solve for [tex]\( r \)[/tex]:
[tex]\[
r = 66 \cdot \frac{7}{44}
\][/tex]
Simplify the fraction inside the multiplication:
[tex]\[
r = 66 \cdot \frac{7}{44} = 66 \cdot \frac{1}{\frac{44}{7}} = 66 \cdot \frac{7}{44}
\][/tex]
Simplify the multiplication by calculating 66 times 7 and then dividing by 44:
[tex]\[
66 \cdot 7 = 462
\][/tex]
[tex]\[
r = \frac{462}{44} = 10.5
\][/tex]
4. Verify the options to find the correct radius:
From the choices given:
[tex]\[
A. \quad 10.5 \quad cm
\][/tex]
[tex]\[
B. \quad 12.5 \quad cm
\][/tex]
[tex]\[
C. \quad 14.0 \quad cm
\][/tex]
[tex]\[
D. \quad 14.5 \quad cm
\][/tex]
[tex]\[
E. \quad 21.0 \quad cm
\][/tex]
Since our calculated radius is [tex]\( 10.5 \)[/tex] cm, the correct option is Option A.
Therefore, the radius of the circle is [tex]\( 10.5 \)[/tex] cm.
