Circle A has a center at #(6 ,2 )# and an area of #45 pi#. Circle B has a center at #(2 ,3 )# and an area of #75 pi#. Do the circles overlap?
yes
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To determine if the circles overlap, we need to compare the distances between their centers to the sum of their radii.
The distance ( d ) between the centers of the circles ( A ) and ( B ) can be calculated using the distance formula:
[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} ]
Given the centers of circle ( A ) and ( B ): ( A: (6, 2) ) ( B: (2, 3) )
Using the distance formula:
[ d = \sqrt{(2 - 6)^2 + (3 - 2)^2} ] [ d = \sqrt{(-4)^2 + (1)^2} ] [ d = \sqrt{16 + 1} ] [ d = \sqrt{17} ]
Now, let's calculate the radii of circles ( A ) and ( B ).
Given the areas of the circles: ( \text{Area}_A = 45\pi ) ( \text{Area}_B = 75\pi )
The radius ( r_A ) of circle ( A ) can be calculated using the formula for the area of a circle: [ \text{Area} = \pi r^2 ]
[ 45\pi = \pi r_A^2 ] [ r_A^2 = \frac{45\pi}{\pi} ] [ r_A^2 = 45 ] [ r_A = \sqrt{45} ]
Similarly, the radius ( r_B ) of circle ( B ) can be calculated: [ 75\pi = \pi r_B^2 ] [ r_B^2 = \frac{75\pi}{\pi} ] [ r_B^2 = 75 ] [ r_B = \sqrt{75} ]
Now, we compare the distance between the centers ( d ) to the sum of the radii ( r_A + r_B ):
[ \sqrt{17} \stackrel{?}{<} \sqrt{45} + \sqrt{75} ]
Since ( \sqrt{17} \approx 4.12 ) and ( \sqrt{45} + \sqrt{75} \approx 9.52 ), we can see that the distance between the centers is less than the sum of the radii. Therefore, the circles do 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.
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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