The center of the original circle can be found from its equation. The equation of the circle given is:
\[ y = \sqrt{16 - (x + 2)^2} - 5 \]
To write the equation in standard form, we can isolate \(y\):
\[ y + 5 = \sqrt{16 - (x + 2)^2} \]
Squaring both sides, we get:
\[ (y + 5)^2 = 16 - (x + 2)^2 \]
From \( (x + 2)^2 + (y + 5)^2 = 16 \), we can identify the center of the circle. The standard form of a circle's equation is:
\[ (x - h)^2 + (y - k)^2 = r^2 \]
where \((h, k)\) is the center and \(r\) is the radius.
From \((x + 2)^2\), we deduce that \(h = -2\).
From \((y + 5)^2\), we deduce that \(k = -5\).
Thus, the center of the original circle is:
\[ (-2, -5) \]
Now, when the circle is translated 3 units to the right, we add 3 to the \(x\)-coordinate of the center:
\[ -2 + 3 = 1 \]
When the circle is translated 3 units up, we add 3 to the \(y\)-coordinate of the center:
\[ -5 + 3 = -2 \]
Therefore, the new center of the translated circle is:
\[ (1, -2) \]
So the answer is:
\[ (1, -2) \]