This modeling exploration follows the exploration seen [url=https://www.geogebra.org/m/ex29ay9z]here[/url]. [br][br]In 2D, consider the equation [math]x^2+y^2=16[/math]. It's graph is a circle with center (0,0) and radius = 4. [br]To see why, [url=https://www.geogebra.org/m/RCYvXnuR]check out this resource here[/url]. [br][br]Yet in this equation, if we replace y with z, this equation becomes [math]x^2+z^2=16[/math]. [br][br]If we solve explicitly for z, we get [math]z=\sqrt{16-x^2}[/math] (upper semicircle) and [math]z=-\sqrt{16-x^2}[/math] (lower semicircle).[br][b]In 3D, think of z as the new DEPENDENT VARIABLE. [/b][br][br]If we graph these 2 surfaces in 3D, the value of y doesn't matter. Thus, these semicircles become infinitely long half cylinders. When they're put together, we get an infinitely long cylinder.[br] [br]What happens when we add the terms (containing y) to the right side of each equation? Why does this occur?[b][i] [br][br]What other kinds of surfaces can we create? [/i][/b]