Circle A has a center at #(5 ,4 )# and a radius of #3 #. 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?
They don't overlap. The smallest distance between them is
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To determine if the circles overlap, we can calculate the distance between the centers of the two circles 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 overlap, and the smallest distance between them is the difference between the distance between their centers and the sum of their radii.
Let's calculate:

The distance between the centers of the circles using the distance formula:
[ d = \sqrt{(x_2  x_1)^2 + (y_2  y_1)^2} ] 
Compare the distance between the centers to the sum of their radii.
Circle A: Center ( (5, 4) ), Radius ( r_A = 3 )
Circle B: Center ( (6, 8) ), Radius ( r_B = 2 )
Calculating the distance between the centers:
[ d = \sqrt{(6  5)^2 + (8  4)^2} ]
[ d = \sqrt{1^2 + (12)^2} ]
[ d = \sqrt{1 + 144} ]
[ d = \sqrt{145} ]
Comparing to the sum of radii:
[ d = \sqrt{145} \approx 12.04 ]
[ r_A + r_B = 3 + 2 = 5 ]
Since ( \sqrt{145} > 5 ), 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:
[ \text{Smallest distance} = \sqrt{145}  5 \approx 7.04 ]
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When evaluating a onesided 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 onesided 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 onesided 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 onesided 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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