Question 11 of 20 This question: 1 point(s) possible Submite Solve the right triangle ABC , with \( \mathrm{C}=90^{\circ} \). \[ \mathrm{A}=57.4^{\circ}, \mathrm{c}=22.4 \mathrm{ft} \] \( B= \) \( \square \) \( { }^{\circ} \) (Simplify your answer. Type an integer or a decimal. Round to the nearest tenth as needed.) \( a= \) \( \square \) ft (Simplify your answer. Type an integer or a decimal. Round to the nearest tenth as needed.) \( b= \) \( \square \) ft (Simplify your answer. Type an integer or a decimal. Round to the nearest tenth as needed.)

We have a right triangle with angle \( C = 90^\circ \), angle \( A = 57.4^\circ \), and the hypotenuse \( c = 22.4 \) ft. The goal is to find angle \( B \), side \( a \) (opposite angle \( A \)), and side \( b \) (opposite angle \( B \)). 1. Since the sum of the angles in a triangle is \( 180^\circ \) and \( C=90^\circ \), we have: \[ A + B + C = 180^\circ \quad \Longrightarrow \quad 57.4^\circ + B + 90^\circ = 180^\circ. \] Solving for \( B \): \[ B = 180^\circ - 90^\circ - 57.4^\circ = 32.6^\circ. \] 2. Use trigonometric ratios to find the missing sides. Since \( c \) is the hypotenuse, we use: - For side \( a \) (opposite angle \( A \)): \[ \sin A = \frac{a}{c} \quad \Longrightarrow \quad a = c \sin A. \] Plug in the known values: \[ a = 22.4 \sin(57.4^\circ). \] Calculating (using a calculator for the sine value): \[ \sin 57.4^\circ \approx 0.8387 \quad \Longrightarrow \quad a \approx 22.4 \times 0.8387 \approx 18.8 \text{ ft}. \] 3. For side \( b \) (opposite angle \( B \)): \[ \sin B = \frac{b}{c} \quad \Longrightarrow \quad b = c \sin B. \] Use the known values: \[ b = 22.4 \sin(32.6^\circ). \] Calculate: \[ \sin 32.6^\circ \approx 0.5394 \quad \Longrightarrow \quad b \approx 22.4 \times 0.5394 \approx 12.1 \text{ ft}. \] Thus, the solutions to the triangle are: \[ B=32.6^\circ, \quad a=18.8 \text{ ft}, \quad b=12.1 \text{ ft}. \]