# Circle A has a center at #(5 ,1 )# and an area of #23 pi#. Circle B has a center at #(2 ,8 )# and an area of #15 pi#. Do the circles overlap?

Yes...

The area of a circle is given by the formula:

Hence:

Then we find:

Since the sum of the radii is greater than the distance between the centres of the circles, they do overlap...

graph{((x-5)^2+(y-1)^2-23)((x-5)^2+(y-1)^2-0.09)((x-2)^2+(y-8)^2-15)((x-2)^2+(y-8)^2-0.1) = 0 [-12, 22, -5, 12]}

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To determine if the circles overlap, calculate the distance between their centers and compare it to the sum of their radii. If the distance between the centers is less than the sum of the radii, the circles overlap; otherwise, they do not.

The distance between two points ((x_1, y_1)) and ((x_2, y_2)) can be found using the distance formula:

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

For Circle A with center ((5, 1)) and Circle B with center ((2, 8)), the distance between their centers is:

[ \text{Distance} = \sqrt{(2 - 5)^2 + (8 - 1)^2} = \sqrt{(-3)^2 + (7)^2} = \sqrt{9 + 49} = \sqrt{58} ]

The radii of Circle A and Circle B can be calculated by taking the square root of their respective areas divided by ( \pi ):

[ \text{Radius of Circle A} = \sqrt{\frac{23\pi}{\pi}} = \sqrt{23} ]

[ \text{Radius of Circle B} = \sqrt{\frac{15\pi}{\pi}} = \sqrt{15} ]

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

[ \sqrt{58} \approx 7.62 ] [ \sqrt{23} + \sqrt{15} \approx 7.62 ]

Since the distance between the centers is equal to the sum of the radii, the circles touch each other at a single point. Therefore, the circles overlap.

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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.

- What is the perimeter of a triangle with corners at #(3 ,0 )#, #(5 ,2 )#, and #(1 ,4 )#?
- Circle A has a center at #(1 ,3 )# and an area of #16 pi#. Circle B has a center at #(2 ,7 )# and an area of #75 pi#. Do the circles overlap?
- What is the perimeter of a triangle with corners at #(7 ,6 )#, #(4 ,5 )#, and #(3 ,1 )#?
- What is the perimeter of a triangle with corners at #(9 ,2 )#, #(2 ,3 )#, and #(4 ,1 )#?
- Circle A has a center at #(5 ,4 )# and a radius of #4 #. Circle B has a center at #(6 ,-8 )# and a radius of #2 #. Do the circles overlap? If not, what is the smallest distance between them?

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