Question

Circle A has a center at `(2 ,3 )` and an area of `8 pi`. Circle B has a center at `(13 ,7 )` and an area of `54 pi`. Do the circles overlap?

Answer 1

This straightforward application of distance formula and comparing of the radiuses.
`R=10.18 < D_((B-A))=11.7` so no overlap

Explanation:

This straightforward application of distance formula and comparing of the radiuses. First let calculate the radii:
Let A(2,3) and B(13,7)
`r_A = pir^2 = 8pi => r=2sqrt(2)`
`r_B = pir^2 = 54pi => r=3sqrt(6)`
so the sum of the radii, `R = r_B+r_A=2sqrt(2)+3sqrt(6)`
`R = sqrt(2)(2+3sqrt(3))`

Now use the distance formula to find the distance from the center of the circles:
`D_((B-A)) = sqrt((13-2)^2+(7-3)^2)=sqrt(137)`

Now `R=10.18 < D_((B-A))=11.7` so no overlap