Question

determine the remaining sides and angles of the triangle abc.
the measure of angle b is
(simplify your answer. type an integer or a decimal.)
the length of side a is ft
(simplify your answer. type an integer or a decimal. round to the nearest tenth as needed.)
the length of side b is ft
(simplify your answer. type an integer or a decimal. round to the nearest tenth as needed.)

Explanation:

Step1: Find angle B

The sum of angles in a triangle is \(180^\circ\). So, \( \angle B = 180^\circ - \angle A - \angle C \). Given \( \angle A = 16.3^\circ \) and \( \angle C = 99.8^\circ \), we substitute these values:
\( \angle B = 180 - 16.3 - 99.8 = 63.9^\circ \)

Step2: Use the Law of Sines to find side a (opposite angle A)

The Law of Sines states \( \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} \), where \( c = 47.6 \) ft (opposite angle C). First, find side a:
\( \frac{a}{\sin 16.3^\circ} = \frac{47.6}{\sin 99.8^\circ} \)
\( a = \frac{47.6 \times \sin 16.3^\circ}{\sin 99.8^\circ} \)
Calculate \( \sin 16.3^\circ \approx 0.2808 \) and \( \sin 99.8^\circ \approx 0.9848 \)
\( a = \frac{47.6 \times 0.2808}{0.9848} \approx \frac{13.366}{0.9848} \approx 13.6 \) ft

Step3: Use the Law of Sines to find side b (opposite angle B)

\( \frac{b}{\sin B} = \frac{47.6}{\sin 99.8^\circ} \)
\( b = \frac{47.6 \times \sin 63.9^\circ}{\sin 99.8^\circ} \)
Calculate \( \sin 63.9^\circ \approx 0.8975 \)
\( b = \frac{47.6 \times 0.8975}{0.9848} \approx \frac{42.721}{0.9848} \approx 43.4 \) ft

Answer:

The measure of angle B is \( 63.9^\circ \)
The length of side a is \( 13.6 \) ft
The length of side b is \( 43.4 \) ft