To find the angle subtended at the center of the hoop by the circular wire, follow these steps:
1. Calculate the circumference of the wire:
Given the radius of the wire is 8 cm, the circumference, [tex]\(C_{\text{wire}}\)[/tex], can be calculated using the formula:
[tex]\[
C = 2 \pi r
\][/tex]
Using [tex]\(\pi = \frac{22}{7}\)[/tex], we get:
[tex]\[
C_{\text{wire}} = 2 \times \frac{22}{7} \times 8 = 50.285714285714285 \, \text{cm}
\][/tex]
2. Calculate the circumference of the hoop:
Given the radius of the hoop is 128 cm, the circumference, [tex]\(C_{\text{hoop}}\)[/tex], can be calculated similarly:
[tex]\[
C_{\text{hoop}} = 2 \times \frac{22}{7} \times 128 = 804.5714285714286 \, \text{cm}
\][/tex]
3. Determine the angle subtended by the wire at the center of the hoop:
The wire now lies along the circumference of the hoop. The angle subtended, [tex]\(\theta\)[/tex], at the center of the hoop can be found using the ratio of the wire's arc length to the hoop's total circumference:
[tex]\[
\theta = \left(\frac{C_{\text{wire}}}{C_{\text{hoop}}}\right) \times 360^\circ
\][/tex]
Substituting the values:
[tex]\[
\theta = \left(\frac{50.285714285714285}{804.5714285714286}\right) \times 360 = 22.5^\circ
\][/tex]
Hence, the angle subtended at the center of the hoop by the circular wire, when bent to lie along the circumference of the hoop, is [tex]\(22.5^\circ\)[/tex].
