Circle C_1 has radius r_1 and Circle C_2 has radius r_2. An arc on Circle C_1 subtended by… — GRE General Test

Circle $C_1$ has radius $r_1$ and Circle $C_2$ has radius $r_2$ . An arc on Circle $C_1$ subtended by a central angle of $\theta_1^\circ$ has the exact same length as an arc on Circle $C_2$ subtended by a central angle of $\theta_2^\circ$ . The sector formed by this arc in Circle $C_1$ has an area of $54\pi$ , and the sector formed by this arc in Circle $C_2$ has an area of $36\pi$ . If $\theta_1 + \theta_2 = 150$ , what is the value of $r_1$ ?

Answer: 18

Answer

The radius

r_1

of Circle

C_1

is 18.

Using the relationship

A = \frac{1}{2} L r

, the ratio of the two sector areas gives

\frac{r_1}{r_2} = \frac{54\pi}{36\pi} = 1.5

, so

r_2 = \frac{2}{3} r_1

. Expressing arc length as

L = \frac{108\pi}{r_1}

, the central angles are

\theta_1 = \frac{19440}{r_1^2}

and

\theta_2 = \frac{29160}{r_1^2}

. Setting their sum equal to

150

leads to

\frac{48600}{r_1^2} = 150

, giving

r_1^2 = 324

and

r_1 = 18

Step-by-Step Solution

Relate sector area to arc length and radius

A_1 = \frac{1}{2} L r_1 = 54\pi

and

A_2 = \frac{1}{2} L r_2 = 36\pi

The area of a sector with arc length

L

and radius

r

is given by

A = \frac{\theta}{360}\pi r^2 = \frac{1}{2} L r

Find the ratio of

r_1

r_2

\frac{r_1}{r_2} = \frac{54\pi}{36\pi} = \frac{3}{2} \implies r_2 = \frac{2}{3}r_1

Dividing the first area equation by the second cancels out

\frac{1}{2}L

Express central angles in terms of

r_1

\theta_1 = \frac{19440}{r_1^2}

and

\theta_2 = \frac{29160}{r_1^2}

Since

L = \frac{108\pi}{r_1}

, substituting into

\theta_1 = \frac{L}{2\pi r_1} \times 360

yields

\theta_1 = \frac{19440}{r_1^2}

, and substituting into

\theta_2 = \frac{L}{2\pi (\frac{2}{3}r_1)} \times 360

yields

\theta_2 = \frac{29160}{r_1^2}

Solve for

r_1

using the angle sum equation

r_1 = 18

Summing the angles gives

\frac{19440 + 29160}{r_1^2} = \frac{48600}{r_1^2} = 150 \implies r_1^2 = 324 \implies r_1 = 18

Key Concept

Relationship between arc length, radius, central angle, and sector area

Estimated Time:2m 30s

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