The equation of a circle is $x^2+(y-10)^2=16$.
The radius of the circle is □ units.
The center of the circle is at □.
The equation of a circle in its standard form is given by the expression \((x - h)^2 + (y - k)^2 = r^2\), where \((h, k)\) represents the coordinates of the center of the circle, and \(r\) represents its radius. Our objective is to identify these values for the given circle equation, which is \(x^2 + (y - 10)^2 = 16\).
We can rewrite the given equation \(x^2 + (y - 10)^2 = 16\) to match the standard form \((x - h)^2 + (y - k)^2 = r^2\).
The term \(x^2\) is equivalent to \((x - 0)^2\), which indicates that the \(x\)-coordinate of the center, \(h\), is \(0\).
Comparing the term \((y - 10)^2\) with \((y - k)^2\), we find that the \(y\)-coordinate of the center, \(k\), is \(10\).
Therefore, the center of the circle \((h, k)\) is \((0, 10)\).
In the standard form \((x - h)^2 + (y - k)^2 = r^2\), the constant on the right side represents the square of the radius, \(r^2\).
Given the equation \(x^2 + (y - 10)^2 = 16\), we have \(r^2 = 16\).
To find the radius \(r\), we take the square root of both sides: \(r = \sqrt{16} = 4\).
Since a radius must be a positive distance, we conclude that the radius is \(4\) units.
Based on the comparison with the standard form, we have determined that the center of the circle is \((0, 10)\) and the radius is \(4\) units.
The final answer is:
The radius of the circle is \(\boxed{4}\) units.
The center of the circle is at \(\boxed{(0, 10)}\).
Circles are fundamental in geometry and various practical fields. For example, if you are planning the layout of a circular fountain in a garden, you might place the center at a specific coordinate \((h, k)\) on your design map. If the fountain's equation is \(x^2 + (y-10)^2 = 16\), you know its center is at \((0, 10)\) and its radius is \(4\) units, allowing you to mark the exact area the water feature will occupy in your garden plan.