The correct answer is option (B): (38 < y < 94)
Let's simplify the expression for y. We are given that
y = (x^2 + 4 * sqrt(x) * (x + 16) + 256) / (x + 8 * sqrt(x) + 16)
The numerator can be rewritten by letting t = √x. Then
y = (t^4 + 4t^3 + 64t + 256) / (t^2 + 8t + 16).
Factor the numerator:
t^4 + 4t^3 + 64t + 256 = (t + 4)^2 (t^2 - 4t + 16)
and the denominator is (t + 4)^2. Canceling the common factor gives
y = t^2 - 4t + 16 = x - 4√x + 16.
Since √x = t ranges from 8 to 11 when x is between 64 and 121,
y(8) = 48 and y(11) = 93, so
48 ≤ y ≤ 93 (or 48 < y < 93 for open endpoints).
Therefore option (B) is correct.
Final Answer: 38 < y < 94