# Circle A has a center at #(4 ,-8 )# and a radius of #3 #. Circle B has a center at #(-2 ,-2 )# and a radius of #2 #. Do the circles overlap? If not, what is the smallest distance between them?

The circles do not overlap and the shortest distance is

The distance between the centers is

The sum of the radii is

As,

The circles do not overlap.

The smallest distance is

graph{((x-4)^2+(y+8)^2-9)((x+2)^2+(y+2)^2-4)(y+x+4)=0 [-25.84, 25.46, -16.57, 9.1]}

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To determine if the circles overlap, we can calculate the distance between their centers and compare it to the sum of their radii. If the distance between the centers is greater than the sum of the radii, the circles do not overlap. Otherwise, they overlap.

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

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

For Circle A with center (4, -8) and Circle B with center (-2, -2), the distance is:

[ \sqrt{(-2 - 4)^2 + (-2 - (-8))^2} = \sqrt{(-6)^2 + (6)^2} = \sqrt{36 + 36} = \sqrt{72} \approx 8.49 ]

The sum of the radii of Circle A and Circle B is (3 + 2 = 5). Since the distance between their centers (( \sqrt{72} )) is greater than the sum of their radii, the circles do not overlap.

The smallest distance between them is the difference between the distance of their centers and the sum of their radii:

[ \text{Smallest distance} = \sqrt{72} - 5 \approx 8.49 - 5 = 3.49 ]

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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- Circle A has a center at #(3 ,2 )# and a radius of #2 #. Circle B has a center at #(1 ,3 )# and a radius of #4 #. Do the circles overlap? If not, what is the smallest distance between them?
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