Final answer:
The curvature of the curve y² = 16x at the point (4, 8) is 1/32, not option D) 16. We find this result by computing the first and second derivatives of y, and then applying the curvature formula at the given point.
Step-by-step explanation:
The curvature of the curve y² = 16x at the point (4, 8) requires a few steps to find. First, we need to express y as a function of x, which will be y = 4√x. Taking the first derivative, we get y' = 2/√x, and substituting the point (4,8) into this derivative gives us y'(4) = 2/2 = 1. We then find the second derivative, which is y'' = -1/(2x√ x), and y''(4) = -1/(2⋅ 4 √ 4) = -1/16. Now we utilize the curvature formula κ = |y''| / (1 + (y')²)^(3/2), and after substituting y'(4) = 1 and y''(4) = -1/16, we find that the curvature at the point (4, 8) is 1/32, which is not option D) 16.