To find the radius of the circle, we need to consider the properties of a rectangle inscribed in a circle.

Given:
- Rectangle ABCD inscribed in a circle with center O.
- AC is the diagonal of the rectangle.
- ∠BAC = 30°.

Properties of a Rectangle Inscribed in a Circle:
1. The diagonal of a rectangle is equal to the diameter of the circle.
2. The diagonals of a rectangle are equal in length and bisect each other at right angles.
3. The diagonals of a rectangle are also the radius of the circle.

Let's solve the problem step by step.

Step 1: Draw the rectangle ABCD and the circle with center O.

O
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A _______ B
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D|_______|C

Step 2: Draw the diagonal AC and label the points where it intersects the circle as E and F.

O
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A _______ B
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D|_______|C
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E F

Step 3: Since ∠BAC = 30°, we know that ∠BCE = 90° - ∠BAC = 60° (angle in a semicircle).

Step 4: In triangle BCE, we have ∠BCE = 60° and ∠ECB = 90° (as BC is a side of the rectangle).

Step 5: Using trigonometric ratios, we can find the relationship between the sides of the triangle BCE.

In a right-angled triangle,
sin θ = Opposite / Hypotenuse
cos θ = Adjacent / Hypotenuse
tan θ = Opposite / Adjacent

In triangle BCE,
sin 60° = BC / BE
cos 60° = CE / BE

Step 6: Simplifying the trigonometric ratios using the values of sin 60° and cos 60°,

√3/2 = BC / BE
1/2 = CE / BE

Step 7: Multiplying both equations by 2,

√3 = (2/BE) * BC
1 = (2/BE) * CE

Step 8: Since CE is the radius of the circle and BE is the radius of the circle (as they are both radii of the same circle), we can simplify the equation as follows:

√3 = (2/r) * BC
1 = (2/r) * r

Step 9: Canceling out the r terms, we get:

√3 = 2 * BC
1 = 2

Step 10: Simplifying further,

BC = √3/2
1 = 1

Therefore, the radius of the circle is equal to BC, which is the length of one side of the rectangle.

Hence, the correct answer is option 'B'.