Two annuluses

The area of the annular circle formed by two circles with a common center is 100 cm². The radius of the outer circle is equal to twice the radius of the inner circle. Determine the outside circle radius in centimeters.

Correct answer:

r₂ =  6.5147 cm

Step-by-step explanation:

$$\begin{gathered} S=100{\text{cm}}^2 \\ {r}_2=2r_1 \\ S=S_2-S_1=\pi \cdot r_2^2-\pi \cdot r_1^2 \\ S/\pi =r_2^2-(r_2/2{)}^2 \\ S/\pi =r_2^2-r_2^2/4 \\ {r}_2=\sqrt{S/\pi /(1-1/4)}=\sqrt{100/3.1416/(1-1/4)}\doteq 6.5147\text{cm} \\ \text{ Verifying Solution: } \\ {r}_1=r_2/2=6.5147/2\doteq 3.2574\text{cm} \\ {S}_3=\pi \cdot r_2^2-\pi \cdot r_1^2=3.1416\cdot 6.5147^2-3.1416\cdot 3.2574^2=100 \\ {S}_3=S \\ {r}_2=6.5147=6.5147\text{cm} \end{gathered}$$

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