Question

Circle A has a center at `(5 ,2 )` and an area of `18 pi`. Circle B has a center at `(3 ,6 )` and an area of `27 pi`. Do the circles overlap?

Answer 1

The circles do overlap (by `4.87` units)

Explanation:

Since Circle A has an Area `=18pi`,
Circle A has a Radius of `r_A = sqrt(18) = 3sqrt(2)~~4.24`
`color(white)("XXX")`(this follows from formua `"Area" =pir^2`)

Since Circle B has an Area `=27pi`
Circle B has a Radius of `r_B=sqrt(27)=3sqrt(3)~~5.20`

The distance between the center of A at `(5,2)` and the center of B at `(3,6)` is
`color(white)("XXX")d=sqrt((5-3)^2+(2-6)^2)=2sqrt(5)~~4.47`

Together the radii of circles A and B cover `4.24+5.20 = 9.34` of the line segment joining the centers of A and B.
Since this is greater than the actual length of the line segment joining A and B, the radii must overlap by `9.34-4.47=4.87` units.
graph{((x-5)^2+(y-2)^2-18)((x-3)^2+(y-6)^2-27)=0 [-14.11, 17.94, -3.74, 12.28]}