To solve the problem, we need to find the value of \( x \) given that an arc of a circle has a length of 22 cm and subtends an angle of \( 3x^\circ \) at the center of the circle. The diameter of the circle is given as 14 cm. ### Step 1: Find the Radius of the Circle The diameter of the circle is 14 cm, so we can find the radius \( r \) using the formula: \[ r = \frac{\text{diameter}}{2} = \frac{14 \text{ cm}}{2} = 7 \text{ cm} \] ### Step 2: Use the Arc Length Formula The formula for the length of an arc \( L \) of a circle is given by: \[ L = \frac{\theta}{360^\circ} \times 2\pi r \] where: - \( L \) is the arc length, - \( \theta \) is the angle in degrees, - \( r \) is the radius of the circle. In this case, we know: - \( L = 22 \text{ cm} \) - \( r = 7 \text{ cm} \) - \( \theta = 3x^\circ \) ### Step 3: Substitute Values into the Arc Length Formula Substituting the known values into the arc length formula, we have: \[ 22 = \frac{3x}{360} \times 2\pi \times 7 \] ### Step 4: Simplify the Equation First, calculate \( 2\pi \times 7 \): \[ 2\pi \times 7 = 14\pi \] Now, substitute this back into the equation: \[ 22 = \frac{3x}{360} \times 14\pi \] ### Step 5: Isolate \( x \) To isolate \( x \), we can rearrange the equation: \[ 22 = \frac{3x \times 14\pi}{360} \] Multiply both sides by \( 360 \): \[ 22 \times 360 = 3x \times 14\pi \] Calculating \( 22 \times 360 \): \[ 7920 = 3x \times 14\pi \] Now, divide both sides by \( 14\pi \): \[ x = \frac{7920}{3 \times 14\pi} \] Calculating \( 3 \times 14 = 42 \): \[ x = \frac{7920}{42\pi} \] ### Step 6: Calculate the Value of \( x \) Now, we can simplify \( \frac{7920}{42} \): \[ 7920 \div 42 = 188.5714 \quad (\text{approximately}) \] Thus, we have: \[ x = \frac{188.5714}{\pi} \] Using \( \pi \approx 3.14 \): \[ x \approx \frac{188.5714}{3.14} \approx 60 \] ### Conclusion Thus, the value of \( x \) is approximately \( 60^\circ \). ### Final Answer The correct option is **A. 60°**. ### Explanation of Other Options - **B. 120°**: This would imply a larger arc length than given, as the angle is too large for the specified arc length. - **C. 180°**: This would correspond to a semicircle, which would yield an arc length of \( 22 \text{ cm} \) only if the radius were larger than 7 cm. - **D. 30°**: This would yield a much smaller arc length than 22 cm, as the angle is too small. ### Revision Summary - The diameter of the circle is 14 cm, giving a radius of 7 cm. - The arc length formula relates the arc length to the angle and radius. - By substituting known values and isolating \( x \), we found \( x \approx 60^\circ \). - The correct answer is option A, as the other options do not satisfy the arc length condition.