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 result:

r2 =  6.515 cm

Solution:

S=100 cm2 r2=2 r1 S=S2−S1=π⋅ r22−π⋅ r12 S/π=r22−(r2/2)2 S/π=r22−r22/4 r2=S/π/(1−1/4)=100/3.1416/(1−1/4)≐6.5147 cm   Correctness test:  r1=r2/2=6.5147/2≐3.2574 cm S3=π⋅ r22−π⋅ r12=3.1416⋅ 6.51472−3.1416⋅ 3.25742=100 S3=S  r2=6.5147=6.515 cmS=100 \ \text{cm}^2 \ \\ r_{2}=2 \ r_{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{ Correctness test: } \ \\ 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=6.515 \ \text{cm}

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