Solve the triangle. \( \mathrm{a}=96.0 \) in., \( \mathrm{b}=60.0 \mathrm{in} ., \mathrm{B}=25.0^{\circ} \) Select the correct choice below and fill in the answer boxes within the choice. Round to the nearest tenth as needed.) A. There is only 1 possible solution for the triangle. The measurements for the remaining angles A and C and side c are as follows. \( \mathrm{A} \approx \square^{\circ} \) \( \mathrm{C} \approx \square^{\circ} \) B . There are 2 possible solutions for the triangle. The measurements for the solution with the longer side c are as follows. \( \mathrm{A} \approx \square^{\circ} \) \( \mathrm{C} \approx \square^{\circ} \) The measurements for the solution with the shorter side c are as follows. \( \mathrm{A} \approx \square^{\circ} \) \( \mathrm{C} \approx \square^{\circ} \) C. There are no possible solutions for this triangle. \( \mathrm{C} \approx \square^{\circ} \) in. in.

To solve the given triangle with \( a = 96.0 \) inches, \( b = 60.0 \) inches, and \( \angle B = 25.0^\circ \), we'll use the **Law of Sines**. This problem falls under the SSA (Side-Side-Angle) scenario, which can result in **0, 1, or 2** possible solutions. ### Step 1: Apply the Law of Sines \[ \frac{\sin A}{a} = \frac{\sin B}{b} \] \[ \sin A = \frac{a \cdot \sin B}{b} = \frac{96.0 \cdot \sin 25.0^\circ}{60.0} \approx \frac{96.0 \cdot 0.4226}{60.0} \approx 0.6762 \] ### Step 2: Determine Possible Angles for \( \angle A \) Since \( \sin A \approx 0.6762 \), there are **two possible angles**: 1. **Primary Solution:** \[ A_1 = \arcsin(0.6762) \approx 42.5^\circ \] 2. **Secondary Solution:** \[ A_2 = 180^\circ - 42.5^\circ = 137.5^\circ \] ### Step 3: Calculate Remaining Angles - **For the Primary Solution (\( A \approx 42.5^\circ \)):** \[ C = 180^\circ - B - A_1 = 180^\circ - 25.0^\circ - 42.5^\circ = 112.5^\circ \] - **For the Secondary Solution (\( A \approx 137.5^\circ \)):** \[ C = 180^\circ - B - A_2 = 180^\circ - 25.0^\circ - 137.5^\circ = 17.5^\circ \] ### Step 4: Determine Side \( c \) for Both Solutions - **Primary Solution (Longer side \( c \)):** \[ c = \frac{a \cdot \sin C}{\sin A} = \frac{96.0 \cdot \sin 112.5^\circ}{\sin 42.5^\circ} \approx 131.1 \text{ inches} \] - **Secondary Solution (Shorter side \( c \)):** \[ c = \frac{a \cdot \sin C}{\sin A} = \frac{96.0 \cdot \sin 17.5^\circ}{\sin 137.5^\circ} \approx 42.7 \text{ inches} \] ### Conclusion There are **two possible solutions** for the triangle. **Answer:** **B. There are 2 possible solutions for the triangle.** The measurements for the solution with the longer side \( c \) are as follows. \( \mathrm{A} \approx 42.5^{\circ} \) \( \mathrm{C} \approx 112.5^{\circ} \) The measurements for the solution with the shorter side \( c \) are as follows. \( \mathrm{A} \approx 137.5^{\circ} \) \( \mathrm{C} \approx 17.5^{\circ} \)