Circle Word Problem

[vanbeersj]

The design of a machine part shows it as a circle represented by x^2 + y^2 = 42.5, with a circular hole represented by x^2 + y^2 + 3.06y - 1.24 = 0 cut out. What is the least distance (in cm) from the edge of the hole to the edge of the machine part?

I know since x^2 + y^2 = r^2 that the radius of the machine part is r = 6.52.

I now have to find the radius of the second equation x^2 + y^2 + 3.06y - 1.24 = 0 which appears to be in the general formula for a circle.
This is my work:

x^2 + (y^2 + 3.06y + 5.79^2) = 5.79^2-1.24
x^2 + (y^2 + 5.79)^2 = 32.28
r^2 = 32.28
r = 5.68

So the distance between the two is 6.52- 5.68 = 0.84 cm.
I just don't know if I've completed the square properly.

 

[wjm11]

The design of a machine part shows it as a circle represented by x^2 + y^2 = 42.5, with a circular hole represented by x^2 + y^2 + 3.06y - 1.24 = 0 cut out. What is the least distance (in cm) from the edge of the hole to the edge of the machine part?

I know since x^2 + y^2 = r^2 that the radius of the machine part is r = 6.52.

I now have to find the radius of the second equation x^2 + y^2 + 3.06y - 1.24 = 0 which appears to be in the general formula for a circle.
This is my work:

x^2 + (y^2 + 3.06y + 5.79^2) = 5.79^2-1.24
x^2 + (y^2 + 5.79)^2 = 32.28
r^2 = 32.28
r = 5.68

So the distance between the two is 6.52- 5.68 = 0.84 cm.
I just don't know if I've completed the square properly.

Two initial comments: you have not completed the square properly, and you cannot simply subtract one radius from the other to find the distance of the hole from the edge.

x^2 + y^2 + 3.06y - 1.24 = 0

x^2 + y^2 + 3.06y = 1.24

To complete the square, divide 3.06 by 2 and square the answer. That amount must be added to both sides of the equation:

x^2 + y^2 + 3.06y + 1.53^ = 1.24 = 1.53^2

x^2 + (y + 1.53)^2 = 3.5809

x^2 + (y + 1.53)^2 = 1.8923^2

Draw a sketch. The machine part is centered on the origin and has a radius of 6.5192. The cutout is centered at (0, -1.53) [on the y-axis, below the origin] and has a radius of 1.8923. The lowest point on the cutout is at (0, -3.4223). The lowest point on the machine part is at (0, -6.5192). The distance between these points is

6.5192 – 3.4223 = 3.0969 (approx.)