Circle A has a center at #(2 ,8 )# and an area of #8 pi#. Circle B has a center at #(3 ,2 )# and an area of #27 pi#. Do the circles overlap?

Applying that in this case, we get the distance between the centers of the circle to be

As a matter of estimation, we can tell that

and

If we actually calculate the values, we get

meaning the estimation was correct, and the circles do overlap.

To determine if the circles overlap, we need to compare the distance between their centers to the sum of their radii. If the distance between the centers is less than the sum of their radii, the circles overlap.

The distance between the centers of Circle A and Circle B can be found using the distance formula:

[ \text{Distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} ]

For Circle A: ( (x_1, y_1) = (2, 8) ) For Circle B: ( (x_2, y_2) = (3, 2) )

[ \text{Distance} = \sqrt{(3 - 2)^2 + (2 - 8)^2} ] [ \text{Distance} = \sqrt{1 + 36} ] [ \text{Distance} = \sqrt{37} ]

The radius of Circle A can be found by taking the square root of its area divided by pi:

[ \text{Radius of Circle A} = \sqrt{\frac{8 \pi}{\pi}} ] [ \text{Radius of Circle A} = \sqrt{8} ]

The radius of Circle B can be found similarly:

[ \text{Radius of Circle B} = \sqrt{\frac{27 \pi}{\pi}} ] [ \text{Radius of Circle B} = \sqrt{27} ]

Now, we compare the distance between the centers to the sum of their radii:

[ \sqrt{37} < \sqrt{8} + \sqrt{27} ]

[ \sqrt{37} < \sqrt{8} + \sqrt{27} ]

[ \sqrt{37} < \sqrt{8} + \sqrt{27} ]

[ \sqrt{37} < \sqrt{8} + \sqrt{27} ]

[ \sqrt{37} < 2.83 + 5.2 ]

[ \sqrt{37} < 8.03 ]

Since the distance between the centers (( \sqrt{37} )) is less than the sum of their radii (8.03), the circles overlap.