Final answer:
The standard equation of the ellipse with center at the origin, foci at (4, 0), and vertices at (5, 0) is x^2/16 + y^2/4 = 1.
Step-by-step explanation:
To write the standard equation of an ellipse, we need to know the coordinates of its center, the lengths of its major and minor axes, and the orientation of its axes. In this case, the center of the ellipse is at the origin (0,0) and the major axis (2a) is of length 10 (since the vertices are at (5,0) and (-5,0)). The distance between the center and one of the foci is also a=4 (since the foci are at (4,0) and (-4,0)).
Since the foci and center are on the x-axis, the equation of the ellipse is x^2/a^2 + y^2/b^2 = 1 where b is the length of the minor axis. To find b, we can use the relationship b^2 = a^2 - c^2, where c is the distance between the center and one of the foci. So in this case, b^2 = 4^2 - 4^2, which gives us b = 2.
Therefore, the standard equation of the ellipse is x^2/16 + y^2/4 = 1.