# Circle A has a center at #(1 ,3 )# and an area of #45 pi#. Circle B has a center at #(2 ,7 )# and an area of #75 pi#. Do the circles overlap?

Two circles overlap

Since the radii of the two circles are greater than the distance between the centers of the two circles, circles overlap.

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To determine if the circles overlap, we can compare the distance between their centers 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 do overlap.

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

Distance = sqrt((x2 - x1)^2 + (y2 - y1)^2)

Using the given coordinates: For Circle A: Center at (1, 3) For Circle B: Center at (2, 7)

Distance = sqrt((2 - 1)^2 + (7 - 3)^2) = sqrt(1 + 16) = sqrt(17)

The radii of the circles can be found using the formula for the area of a circle:

Area = π * radius^2

For Circle A: 45π = π * radius^2 => radius_A = sqrt(45) = 3sqrt(5) For Circle B: 75π = π * radius^2 => radius_B = sqrt(75) = 5sqrt(3)

The sum of the radii is radius_A + radius_B = 3sqrt(5) + 5sqrt(3).

Comparing the distance between the centers (sqrt(17)) to the sum of the radii (3sqrt(5) + 5sqrt(3)), we see that sqrt(17) is less than 3sqrt(5) + 5sqrt(3). Hence, 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.

- Given #L_1->x+3y=0#, #L_2=3x+y+8=0# and #C_1=x^2+y^2-10x-6y+30=0#, determine #C->(x-x_0)^2+(y-y_0)^2-r^2=0# tangent to #L_1,L_2# and #C_1#?
- A triangle has corners at #(6 ,7 )#, #(2 ,1 )#, and #(5 ,8 )#. How far is the triangle's centroid from the origin?
- A triangle has corners at #(9 ,3 )#, #(6 ,7 )#, and #(3 ,2 )#. How far is the triangle's centroid from the origin?
- Circle A has a center at #(2 ,5 )# and a radius of #2 #. Circle B has a center at #(4 ,2 )# and a radius of #3 #. Do the circles overlap? If not, what is the smallest distance between them?
- Circle A has a center at #(-2 ,-7 )# and a radius of #2 #. Circle B has a center at #(-3 ,2 )# and a radius of #5 #. Do the circles overlap? If not, what is the smallest distance between them?

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