Question

Circle A has a center at `(1 ,3 )` and an area of `24 pi`. Circle B has a center at `(8 ,2 )` and an area of `66 pi`. Do the circles overlap?

Answer 1

Yes

Explanation:

`A_1 = 24pi = pi r_1^2`
`r_1^2 = 24` therefore `r_1 = sqrt(24) = sqrt(6*4) = 2sqrt(6) ~~ 4.9`

`A_2 = 66pi = pi r_2^2`
`r_2^2 = 66` therefore `r_2 = sqrt(66) ~~8.12`

From `(1,3)` add `r_1` to the `x` gives `(5.9, 3)`

From `(8,2)` subtract `r_2` from the `x` gives `(-.12, 2)`

Since `-.12 < 5.9` the circles overlap.

Graph of first circle: `(x-1)^2 + (y-3)^2 = 24`:
graph{(x-1)^2 + (y-3)^2 = 24 [-11.66, 13.65, -3.34, 9.32]}

Graph of second circle: `(x-8)^2 + (y-2)^2 = 66`:
graph{(x-8)^2 + (y-2)^2 = 66 [-15.68, 20.35, -6.93, 11.09]}