Questions: Find the center and radius of the sphere. (x+1)^2+y^2+(z-1)^2=20

Solution Steps

To find the center and radius of the sphere given by the equation \((x+1)^{2}+y^{2}+(z-1)^{2}=20\), we can compare it to the standard form of a sphere's equation \((x-h)^2 + (y-k)^2 + (z-l)^2 = r^2\). Here, \((h, k, l)\) is the center and \(r\) is the radius.

1. Identify the center \((h, k, l)\) by comparing the given equation to the standard form.
2. Determine the radius \(r\) by taking the square root of the constant term on the right side of the equation.

The given equation of the sphere is: \[ (x+1)^{2} + y^{2} + (z-1)^{2} = 20 \] By comparing this with the standard form of a sphere's equation: \[ (x-h)^{2} + (y-k)^{2} + (z-l)^{2} = r^{2} \] we can identify the center \((h, k, l)\) of the sphere. Here, \(h = -1\), \(k = 0\), and \(l = 1\).

The right side of the given equation is 20, which represents \(r^{2}\). To find the radius \(r\), we take the square root of 20: \[ r = \sqrt{20} \approx 4.4721 \]

Final Answer