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13 of 2113 of 21 Items Question A builder is installing a circular window and needs to determine the y-coordinate of the center for proper vertical placement. The window's boundary is described by the equation x2+y2−6x+8y=0 . What is the radius of the window?(1 point) Responses r=9 r=9 r=16 r=16 r=5 r=5 r=25 r=25 Skip to navigation page 1 of 1 Formula keypad has been closed. Press Control + Backslash to open it again.Formula keypad has been closed. Press Control + Backslash to open it again.Formula keypad has been closed. Press Control + Backslash to open it again.Formula keypad has been closed. Press Control + Backslash to open it again.
Answers
GPT-4o mini
To find the radius of the circular window given the equation \( x^2 + y^2 - 6x + 8y = 0 \), we can rearrange this equation into the standard form of a circle, which is:
\[
(x - h)^2 + (y - k)^2 = r^2
\]
where \((h, k)\) is the center of the circle and \(r\) is the radius.
Let's start by rewriting the equation:
1. Rearrange the given equation:
\[
x^2 - 6x + y^2 + 8y = 0
\]
2. Complete the square for \(x\) and \(y\).
- For \(x^2 - 6x\):
- Take \(-6\), divide by \(2\) to get \(-3\), and square it to get \(9\).
- Rewrite:
\[
x^2 - 6x = (x - 3)^2 - 9
\]
- For \(y^2 + 8y\):
- Take \(8\), divide by \(2\) to get \(4\), and square it to get \(16\).
- Rewrite:
\[
y^2 + 8y = (y + 4)^2 - 16
\]
3. Substitute these back into the equation:
\[
(x - 3)^2 - 9 + (y + 4)^2 - 16 = 0
\]
4. Combine constants:
\[
(x - 3)^2 + (y + 4)^2 - 25 = 0
\]
5. Rearrange to the standard form:
\[
(x - 3)^2 + (y + 4)^2 = 25
\]
Now we can identify the center and the radius:
- The center of the window (circle) is at \((3, -4)\).
- The radius \(r\) is given by the square root of \(25\):
\[
r = \sqrt{25} = 5
\]
Thus, the radius of the window is \(5\).
**Final answer:**
\(r = 5\)
\[
(x - h)^2 + (y - k)^2 = r^2
\]
where \((h, k)\) is the center of the circle and \(r\) is the radius.
Let's start by rewriting the equation:
1. Rearrange the given equation:
\[
x^2 - 6x + y^2 + 8y = 0
\]
2. Complete the square for \(x\) and \(y\).
- For \(x^2 - 6x\):
- Take \(-6\), divide by \(2\) to get \(-3\), and square it to get \(9\).
- Rewrite:
\[
x^2 - 6x = (x - 3)^2 - 9
\]
- For \(y^2 + 8y\):
- Take \(8\), divide by \(2\) to get \(4\), and square it to get \(16\).
- Rewrite:
\[
y^2 + 8y = (y + 4)^2 - 16
\]
3. Substitute these back into the equation:
\[
(x - 3)^2 - 9 + (y + 4)^2 - 16 = 0
\]
4. Combine constants:
\[
(x - 3)^2 + (y + 4)^2 - 25 = 0
\]
5. Rearrange to the standard form:
\[
(x - 3)^2 + (y + 4)^2 = 25
\]
Now we can identify the center and the radius:
- The center of the window (circle) is at \((3, -4)\).
- The radius \(r\) is given by the square root of \(25\):
\[
r = \sqrt{25} = 5
\]
Thus, the radius of the window is \(5\).
**Final answer:**
\(r = 5\)