Find the equation of the circle with a center at \( (0,-4) \) and a radius of 7 \( \begin{array}{l}x^{2}+(y+4)^{2}=7 \\ x^{2}-(y+4)^{2}=7 \\ x^{2}+(y+4)^{2}=49 \\ x^{2}+(y-4)^{2}=49\end{array} \)

To find the equation of a circle, we use the standard form, which is given by \( (x - h)^2 + (y - k)^2 = r^2 \), where \( (h, k) \) is the center of the circle and \( r \) is the radius. In this case, the center is \( (0, -4) \) and the radius is \( 7 \). Plugging in the values, we get: \[ (x - 0)^2 + (y + 4)^2 = 7^2 \] This simplifies to: \[ x^2 + (y + 4)^2 = 49 \] So, the correct equation of the circle is \( x^2 + (y + 4)^2 = 49 \). Did you know that circles have been studied since ancient times? The ancient Greeks, particularly mathematicians like Euclid and Apollonius, not only defined a circle as the set of points equidistant from a center but also explored their properties and relationships with other geometric figures in great depth! As for real-world applications, circles pop up everywhere! From designing wheels to the layout of parks and roundabouts, the circle's symmetrical properties make them practical in engineering and architecture. So, the next time you see a circular fountain or a bike tire, you’ll know circles are not just for math class – they’re part of the real world!