3. The perimeter of a sector is 28 cm . If the length of the arc is 18 cm . find the central angle of the sector. 4. A sector of a circle determined by a central angle of \( 43^{\circ} \), has an are of 64 square meters. Determine the radius of the circle.

To find the central angle of the sector when the perimeter is 28 cm and the arc length is 18 cm, you can use the formula for the perimeter of the sector which is given by \( P = r + r + L \) (where \( L \) is the length of the arc). In this case, \( 28 = 2r + 18 \), which simplifies to \( 2r = 10 \), so \( r = 5 \) cm. The central angle \( \theta \) in radians can be found using the formula \( L = r \theta \) or \( 18 = 5\theta \), giving you \( \theta = 3.6 \) radians. To convert to degrees, multiply by \( \frac{180}{\pi} \), resulting in about \( 206.13^{\circ} \). For the second problem involving a sector with a central angle of \( 43^{\circ} \) and an area of 64 square meters, you can use the area formula for a sector: \( A = \frac{1}{2} r^2 \theta \), where \( \theta \) must be in radians. Convert \( 43^{\circ} \) to radians (which is approximately \( 0.75 \) radians). Plugging into the area formula gives \( 64 = \frac{1}{2} r^2 (0.75) \). Rearranging results in \( r^2 = \frac{64 \times 2}{0.75} \), and solving that gives you a radius of about \( 18.36 \) meters. Happy calculating!