Solve the triangle, if possible. \[ \mathrm{a}=200 \mathrm{~m}, \mathrm{~A}=39.02^{\circ}, \mathrm{C}=28.75^{\circ} \] Select the correct choice below and, if necessary, fill in the answer boxes to complete your choice. (Round side lengths to the nearest whole number and angle measures to the nearest hundredth degree as needed.) A. There is only one possible solution for the triangle. The measurements for the remaining angle \( B \) and sides \( b \) and \( c \) are as follows. \( B \approx \) \( \square \) \( { }^{\circ} \) \( b= \) \( \square \) m \[ c= \] \( \square \) m B. There are two possible sốutions for the triangle. The measurements for the solution with the smaller angle \( B \) are as follows. \( \mathrm{B}_{1} \approx \) \( \square \) \( { }^{\circ} \) \[ b_{1}= \] \( \square \) m \[ c_{1}= \] \( \square \) m The measurements for the solution with the larger angle \( B \) are as follows. \( \mathrm{B}_{2} \approx \) \( \square \) - \[ \mathrm{b}_{2}= \] \( \square \) m \[ c_{2}=\square \] m C. There are no possible solutions for this triangle.

**Step 1. Find the remaining angle \( B \):** We know that the sum of angles in any triangle is \(180^\circ\). Thus, \[ B = 180^\circ - (A + C) = 180^\circ - (39.02^\circ + 28.75^\circ) = 180^\circ - 67.77^\circ = 112.23^\circ. \] **Step 2. Use the Law of Sines to determine the remaining sides:** The Law of Sines states that \[ \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}. \] Given \(a = 200\) m and angle \(A = 39.02^\circ\), compute the common ratio \( k \): \[ k = \frac{a}{\sin A} = \frac{200}{\sin 39.02^\circ}. \] Using a calculator, \(\sin 39.02^\circ \approx 0.6293\), so: \[ k \approx \frac{200}{0.6293} \approx 317.8 \text{ m}. \] **Step 3. Compute side \( b \):** \[ b = k \sin B. \] Since \(B \approx 112.23^\circ\) and \(\sin 112.23^\circ = \sin (180^\circ - 112.23^\circ) = \sin 67.77^\circ \approx 0.9238\), we have: \[ b \approx 317.8 \times 0.9238 \approx 293.4 \text{ m}. \] Rounded to the nearest whole number: \[ b \approx 293 \text{ m}. \] **Step 4. Compute side \( c \):** \[ c = k \sin C. \] Given \(C = 28.75^\circ\) and \(\sin 28.75^\circ \approx 0.4818\), we find: \[ c \approx 317.8 \times 0.4818 \approx 152.9 \text{ m}. \] Rounded to the nearest whole number: \[ c \approx 153 \text{ m}. \] **Conclusion:** There is only one possible solution for the triangle with the following measurements: - \( B \approx 112.23^\circ \) - \( b \approx 293 \) m - \( c \approx 153 \) m Thus, the correct choice is **A**.