Final answer:
The nearest point to the origin on the surface defined by y = (x - 1)^(3/2) is (1, 0), as it is the only option given that satisfies the surface's equation and has the minimum distance to the origin when applying the distance formula.
Step-by-step explanation:
To determine the nearest point to the origin in three-dimensional space for a surface defined by the equation y = (x - 1)3/2, we need to find the point on the surface that has the minimum distance to the origin (0, 0). Since the equation given defines a surface in three dimensions, we should consider the z-coordinate as well, but as it's not given, we will work with the 2D projection onto the x-y plane.
Firstly, let's consider the options given: Points (1, 0), (0, 0), (1, 1), and (1, -1). We can immediately rule out (0, 0) since it doesn't satisfy the given equation for y, and (1, 1) and (1, -1) as their y-values do not match when plugged into the equation. The only point that satisfies the given equation is (1, 0), as plugging x = 1 into the equation gives y = (1 - 1)3/2 = 0, which matches the y-coordinate.
The distance formula in two dimensions is √(x2 - x1)2 + (y2 - y1)2. Applying this formula, we find that the distance from the origin to point (1, 0) is √(1 - 0)2 + (0 - 0)2 = √1 = 1, which makes it the closest point on the surface to the origin. Therefore, the correct answer is (a) (1, 0).