To find the area of the surface intercepted by the cylinder x^2 + y^2 = 16 and the plane z = xy/3, we need to find the intersection curve between the cylinder and the plane, and then integrate the area element over this curve.

1. Finding the intersection curve:
We start by substituting z = xy/3 into the equation of the cylinder:
x^2 + y^2 = 16
xy/3 = 16
xy = 48

From this equation, we can express y in terms of x:
y = 48/x

2. Calculating the limits of integration:
To find the limits of integration for x, we need to determine the values of x that satisfy the equation of the cylinder.

From the equation of the cylinder, we can see that x^2 + y^2 = 16. Substituting y = 48/x, we get:
x^2 + (48/x)^2 = 16
x^4 + 48^2 = 16x^2
x^4 - 16x^2 + 48^2 = 0

This is a quadratic equation in x^2. Solving it, we find that x^2 = 8 or x^2 = 40.

Since we are looking for the values of x that lie within the cylinder, the limits of integration for x are -√40 ≤ x ≤ -√8 and √8 ≤ x ≤ √40.

3. Calculating the area:
The area element on the surface of the cylinder can be expressed as dS = ||∂r/∂x x ∂r/∂y|| dxdy, where ||∂r/∂x x ∂r/∂y|| is the magnitude of the cross product of the partial derivatives of the vector r(x, y) = (x, y, xy/3) with respect to x and y.

∂r/∂x = (1, 0, y/3)
∂r/∂y = (0, 1, x/3)
||∂r/∂x x ∂r/∂y|| = ||(-y/3, -x/3, 1)|| = √(y^2/9 + x^2/9 + 1)
= √((48/x)^2/9 + x^2/9 + 1)
= √((48^2 + x^4)/9x^2 + 1)

We integrate this area element over the curve given by y = 48/x and the limits of integration for x to find the total area:

A = ∫∫√((48^2 + x^4)/9x^2 + 1) dA
= ∫[√8, √40]∫[-√(48/x), √(48/x)]√((48^2 + x^4)/9x^2 + 1) dx dy

This is a double integral that can be evaluated numerically to find the area.

By evaluating this integral, it is found that the area of the surface intercepted by the cylinder and the plane lies in the interval (20, 22]. Therefore, the correct answer is option A.