SOLUTION
Given the question in the image, the following are the solution steps to answer the question.
STEP 1: Write the formula for area of a sector
[tex]Areaofsector=\frac{\theta}{360}\times\pi r^2[/tex]
STEP 2: Write the given measures
[tex]\begin{gathered} For\text{ Sector }COD \\ \theta=90\degree \\ radius(r)=? \\ Area\text{ }of\text{ }sector=50.24 \end{gathered}[/tex]
STEP 3: substitute the values in the formula in step 1
[tex]50.24=\frac{90}{360}\times3.14\times r^2[/tex]
Solve for radius(r)
[tex]\begin{gathered} 50.24=\frac{90}{360}\times3.14r^{2} \\ 50.24=\frac{282.6r^2}{360} \\ \\ Cross\text{ Multiply} \\ 50.24\times360=282.6r^2 \\ r^2=\frac{18086.4}{282.6} \\ r^2=64 \\ r=\sqrt{64}=8\text{ unit} \end{gathered}[/tex]
Therefore, the radius of the circle is 8 units
STEP 4: Calculate the measure of arc AB
[tex]length\text{ }of\text{ }an\text{ }arc=\frac{\theta}{360}\times2\pi r[/tex]
STEP 5: Write the known values
[tex]\begin{gathered} \theta=30\degree \\ r=8 \\ \pi=3.14 \end{gathered}[/tex]
STEP 6: calculate the length of the arc
By substitution in to the formula in step 4, we have:
[tex]\begin{gathered} \frac{30}{360}\times2\times3.14\times8=\frac{1507.2}{360}=4.186666667 \\ length\approx4.2units \end{gathered}[/tex]
Hence, the measure of arc AB is approximately 4.2 units
