In triangle ABC, where angle C is the right angle, and angle B is given as 30 degrees, we can use the properties of a right triangle.
Given:
- \( BC = 11 \) (the side opposite angle B)
- \( \angle B = 30^\circ \)
To find \( AC \) (the side adjacent to angle B), we can use the trigonometric function of cosine:
\[ \cos(\angle B) = \frac{\text{adjacent side}}{\text{hypotenuse}} = \frac{AC}{AB} \]
Since we want to find \( AC \) and we know the length of \( BC \) (the side opposite angle B), we can use the sine function:
\[ \sin(\angle B) = \frac{\text{opposite side}}{\text{hypotenuse}} = \frac{BC}{AB} \]
Calculating \( AB \) using \( \sin(30^\circ) = \frac{1}{2} \):
\[ \sin(30^\circ) = \frac{BC}{AB} \implies \frac{1}{2} = \frac{11}{AB} \]
From this, we can solve for \( AB \):
\[ AB = 11 \times 2 = 22 \]
Now we will find \( AC \) using \( \cos(30^\circ) \):
\[ \cos(30^\circ) = \frac{\sqrt{3}}{2} \]
So we have:
\[ \cos(30^\circ) = \frac{AC}{AB} \implies \frac{\sqrt{3}}{2} = \frac{AC}{22} \]
Now we can solve for \( AC \):
\[ AC = 22 \times \frac{\sqrt{3}}{2} = 11\sqrt{3} \]
Thus, the answer for \( AC \) is given by:
\[ \boxed{11\sqrt{3}} \]
Looking at the multiple-choice answers available, the expression for \( AC \) matches:
\[ \frac{11 \sqrt{3}}{2} \quad \text{(the calculation previously is incorrect, however, the correct calculation gives)} \quad 11 \sqrt{3} \text{ (matching none)}, \text{ please note that an interpretation may yield accountable parts of evaluations in answers)} \]
So realize the approach awaits clarification for deriving segments properly, and updating expectations as angles balance across hypotheses in triangles.