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

smallest distance =

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The distance between the centers of Circle A and Circle B can be calculated using the distance formula. It is ( \sqrt{(1 - (-4))^2 + (3 - (-1))^2} = \sqrt{5^2 + 4^2} = \sqrt{25 + 16} = \sqrt{41} \approx 6.4 ) units. Since the sum of the radii of the two circles is ( 3 + 2 = 5 ) units, which is less than the distance between their centers, the circles do not overlap. The smallest distance between them is the difference between the distance between their centers and the sum of their radii, which is ( \sqrt{41} - 5 ) units. Therefore, the smallest distance between the circles is approximately ( 6.4 - 5 = 1.4 ) units.

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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.

- Circle A has a center at #(1 ,8 )# and an area of #15 pi#. Circle B has a center at #(5 ,3 )# and an area of #24 pi#. Do the circles overlap?
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- How many values of x between 0.01 and 1 does the graph #sin(1/x)# cross the x-axis?
- A triangle has corners at #(9 ,1 )#, #(2 ,4 )#, and #(5 ,8 )#. How far is the triangle's centroid from the origin?
- Circle A has a center at #(12 ,2 )# and an area of #13 pi#. Circle B has a center at #(3 ,6 )# and an area of #28 pi#. Do the circles overlap?

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