Question

The circular bull’s-eye of an archery target has a diameter of 24 cm, which is surrounded by 4 concentric rings with a width of 12 cm each. Draw the target in a coordinate plane with center at the origin. Write the equations of the circles that form the boundaries of the different regions of the target.

Answer 1

It appears that the bull's eye is the center of the target and it has a diameter of 24 cm and a center at (0,0). Do you know the standard form for the equation of a circle?

Answer 2

\[(x-h)^{2}+(y-k)^{2}=r ^{2}\]

Answer 3

That's right. In your case, what is the values for the h and the k?

Answer 4

I'm not sure?

Answer 5

Your problem tells you. The bull's eye is centered at the origin. The origin has coordinates of (0,0). So your h is a 0, and your k is a 0. That means that the center of the circle is right at the place where the x and y axes cross. So far your equation is this:
x^2 + y^2 =

Answer 6

What is the radius of the bull's eye?

Answer 7

24 cm

Answer 8

24 cm is the diameter. What is the radius?

Answer 9

Oh, 12 cm

Answer 10

Right! So can you finish the equation for the bull's eye of the target?
x^2 + y^2 = what?

Answer 11

it's r^2 right? Which is the same as the diameter, which is 24 cm

Answer 12

No it's not the same. The radius is half the diameter. 12 * 2 = 24. 12^2 is a completely different thing. That's the same as 12 * 12. What's 12 * 12?

Answer 13

12*12=144

Answer 14

Right! So the equation for the smallest circle is x^2 + y^2 = 144

Answer 15

Now onto the circle the next size up!

Answer 16

That's the bull's eye and the next size up circle. It has a width of 12, which I drew in, but how big is the radius now? Keep in mind that the smaller circle is now a part of the bigger circle. What is the radius of both circles together?

Answer 17

24?

Answer 18

That's right! Can you write the equation for that circle? Remember that it's still centered at the origin, and the bigger circle's radius is 24.

Answer 19

Can we just use the old equation of x^2+y^2=144 and make it x^2+y^2=144^2?

Answer 20

No because 144 squared is 20,736, and the radius of your circle is 24, and 24 squared is 576. See why you can't do that?

Answer 21

Yeah, I see it now. So what do I do?

Answer 22

The equation for the next size up circle is still the x^2 + y^2, but your radius squared is 576. So the equation is x^2 + y^2 = 576.

Answer 23

Onto the next size up circle! There are 4 circles around the bull's eye and this one now is the second one.

Answer 24

Is the equation for the next circle x^2+y^2=1296?

Answer 25

Yes! What about the third one? The radius will be 12 * 4 which is 48. What is the equation?

Answer 26

2304

Answer 27

ok....one more to go. 12*5 = 60. Whats the equation?

Answer 28

x^2+y^2=3600

Answer 29

You got it! Good job! You got all of them now!

Answer 30

Thanks! Can you help with another problem?

Answer 31

sure!

Answer 32

I'll make a new post

Answer 33

Hey @IMStuck how do I draw this on a coordinate plane?