Sure! Let's solve the problem step-by-step:
1. Identify the Given Data:
- The total area of the circle is \(\( 187 \)\) square inches.
- The area of the sector is \(\( 67 \)\) square inches.
2. Understand the Relationship:
- The formula for the area of a sector of a circle is given by:
\(\[
\text{Sector Area} = \left(\frac{\theta}{360}\right) \times \pi r^2
\]\)
where \(\(\theta\)\) is the central angle of the sector in degrees, and \(\(r\)\) is the radius of the circle.
- We are given the total area of the circle, from which we can derive:
\(\[
\text{Total Area} = \pi r^2
\]\)
- Thus, the ratio of the sector area to the total area can also be represented as:
\(\[
\frac{\text{Sector Area}}{\text{Total Area}} = \frac{\theta}{360}
\]\)
3. Apply the Known Values:
- Given \(\(\text{Sector Area} = 67\)\) square inches and \(\(\text{Total Area} = 187\)\) square inches, we substitute these into the ratio:
\(\[
\frac{67}{187} = \frac{\theta}{360}
\]\)
4. Solve for \(\(\theta\)\):
- To isolate \(\(\theta\)\), multiply both sides by \(\(360\)\):
\(\[
\theta = \frac{67}{187} \times 360
\]\)
5. Calculate the Central Angle:
- When the calculation is performed:
\(\[
\theta \approx 128.98^\circ
\]\)
6. Compare to Given Options:
- The obtained angle, approximately \(\(128.98^\circ\)\), needs to be compared against the provided options:
- (1) 60°
- (2) 90°
- (3) 120°
- (4) 240°
Since \(\(128.98^\circ\)\) is closest to one of the given options but does not directly match any, it suggests that none of the provided options is a good fit. This indicates you might verify or cross-check the given options. However, given the angle is about \(\(129^\circ\)\), we see it clearly falls near neither of the choices (1) 60°, (2) 90°, (3) 120°, nor (4) 240°.
Therefore, with the result showing approximately \(\(128.98^\circ\)\), it proves that no exact match is present in the options provided. More precisely, there might have been a problem in listing the closest angles/choices. Double-check on the question approach or choices verification might be needed for refined exactness if applicable precisely!
