Question 10 (1 point)
In triangle $A B C, B=30^{\circ}$ and $c=20$. Two triangles will be formed if $x< b< 20$. The value of $x$ is

Step 1 ：Given a triangle ABC with angle B = 30 degrees and side c = 20. We need to find the value of x such that x < b < 20, where b is the side opposite angle B.

Step 2 ：Using the sine rule: \(\frac{\sin(B)}{b} = \frac{\sin(C)}{c}\)

Step 3 ：Since \(\sin(30) = \frac{1}{2}\), we can rewrite the equation as: \(\frac{1}{2b} = \frac{\sin(C)}{20}\)

Step 4 ：We know that the sum of angles in a triangle is 180 degrees, so: \(A + B + C = 180\) and \(A + 30 + C = 180\), which gives us \(A + C = 150\)

Step 5 ：Using the sine rule again: \(\frac{\sin(A)}{a} = \frac{\sin(C)}{c}\), we can rewrite this equation as: \(\sin(C) = \frac{20 \sin(A)}{a}\)

Step 6 ：Substitute this expression for sin(C) back into our original equation: \(\frac{1}{2b} = \frac{20 \sin(A)}{20a}\), which simplifies to \(\frac{1}{2b} = \frac{\sin(A)}{a}\)

Step 7 ：Solve for b in terms of sin(A) and a: \(b = 2a \sin(A)\)

Step 8 ：Find the minimum value of b when sin(A) is at its maximum value of 1 and a is at its minimum value. The minimum value of a will occur when angle A is at its minimum value. Since A + C = 150, the minimum value of angle A will occur when angle C is at its maximum value of 180 - 30 = 150 degrees. In this case, angle A will be 0 degrees, which is not possible in a triangle. Therefore, we need to consider the next smallest angle for A, which is 1 degree. In this case, angle C will be 149 degrees.

Step 9 ：Plug in these values for angle A and a into our equation for b: \(b = 2a \sin(A)\), which gives us \(b = 2a \sin(1)\)

Step 10 ：Using Python code, we find that the minimum value of b is approximately 0.0349.

Step 11 ：\(\boxed{x \approx 0.0349}\)