# 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

By signing up, you agree to our Terms of Service and Privacy Policy

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.

By signing up, you agree to our Terms of Service and Privacy Policy

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 #(6 ,5 )#, #(8 ,2 )#, and #(4 ,7 )#?
- A triangle has corners at #(1 ,4 )#, #(7 ,6 )#, and #(4 ,5 )#. How far is the triangle's centroid from the origin?
- The North Campground (3,5) is midway between the North Point Overlook (1,y) and the Waterfall (x,1). How do I use the Midpoint Formula to find the values of x and y and justify each step? Please show steps.
- What is the perimeter of a triangle with corners at #(3 ,6 )#, #(1 ,5 )#, and #(8 ,1 )#?
- A triangle has corners at #(7 ,9 )#, #(1 ,4 )#, and #(3 ,8 )#. How far is the triangle's centroid from the origin?

- 98% accuracy study help
- Covers math, physics, chemistry, biology, and more
- Step-by-step, in-depth guides
- Readily available 24/7