Question

Two concentric circles with radii of 19 and 29 units bound a shaded region. A third circle will be drawn with area equal to that of the shaded area. What must the radius of the third circle be? Express your answer in simplest radical form.

Answer 1

Answer: 4*sqrt(30)

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Work Shown:

Let P and Q be two concentric circles

P = area of circle with radius 19

P = pi*r^2

P = pi*19^2

P = 361pi

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Q = area of circle with radius 29

Q = pi*r^2

Q = pi*29^2

Q = 841pi

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R = area of shaded region between circle P and circle Q

R = region outside circle P, but inside circle Q

R = Q - P

R = 841pi - 361pi

R = 480pi

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S = area of circle with area equal to region R's area

S = 480pi

S = pi*r^2

pi*r^2 = 480pi

r^2 = 480

r = sqrt(480)

r = sqrt(16*30)

r = sqrt(16)*sqrt(30)

r = 4*sqrt(30)