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A sector of a circle is shown below.  The sector has an area of $60 \pi.$  What is the radius of the circle?

 

 May 4, 2024
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This is a pretty easy problem to solve, set I'll try my best to explain it!

 

We know the area of a circle is \(r^2\pi \). This is when we have the full 360 degrees. However, for this problem, we have the fact that only 83 degrees is shown. 

 

This is easily solvable, as that means that 83/360 of the total area has been calculated. Please understand why this is. 

 

So, we can find the radius with the equation \(\frac{83}{360}r^2\pi = 60\pi\). Canceling out the pi and isolating r^2, we have \(r^2= 60 \cdot \frac{360}{83}\)

 

This is where it gets a bit messy. I won't show all my work, but just know that after canceling everything out, we have \(r^2 = \frac{21600}{83}\). Squaring rooting both sides we have \(r=\frac{60\sqrt{498}}{83}\)

 

Thanks!

 May 4, 2024

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