The area of a sector of central angle \(200^\circ\) of a circle is \(770\,\text{cm}^2\). Find the length of the corresponding arc.
\(\dfrac{70}{3}\,\pi\,\text{cm} \;\approx\; 73.3\,\text{cm}\)
Step 1: Recall the formula for the area of a sector:
\[ A = \dfrac{\theta}{360^\circ} \times \pi r^2 \]
where:
Step 2: Substitute the given values:
Area \(A = 770\,\text{cm}^2, \; \theta = 200^\circ, \; \pi = \dfrac{22}{7}\).
So,
\[ 770 = \dfrac{200}{360} \times \pi r^2 \]
Step 3: Simplify the fraction:
\[ \dfrac{200}{360} = \dfrac{5}{9} \]
So,
\[ 770 = \dfrac{5}{9} \pi r^2 \]
Step 4: Solve for \(r^2\):
\[ r^2 = \dfrac{770 \times 9}{5 \pi} \]
Substitute \(\pi = \dfrac{22}{7}\):
\[ r^2 = \dfrac{6930}{5 \times \tfrac{22}{7}} = 441 \]
So, \(r = \sqrt{441} = 21\,\text{cm}.\)
Step 5: Recall the formula for arc length:
\[ L = \dfrac{\theta}{360^\circ} \times 2 \pi r \]
Step 6: Substitute the values:
\[ L = \dfrac{200}{360} \times 2 \pi \times 21 \]
\[ L = \dfrac{5}{9} \times 42 \pi \]
\[ L = \dfrac{70}{3} \pi \, \text{cm} \]
Step 7: Approximate using \(\pi \approx 3.1416\):
\[ L \approx 73.3\,\text{cm} \]
Final Answer: The length of the arc is \(\dfrac{70}{3}\pi\,\text{cm} \;\approx\; 73.3\,\text{cm}.\)