Question

applying the law of sines to multiple triangles a surveyor found the angle of elevation from the ground to the top of the building at two locations 20 feet apart as shown. which measurements are correct? round side lengths to the nearest hundredth. choose two correct answers. ( mangle b = 15^circ ) ( mangle a = 57^circ ) ( h approx 8.09\text{ft} )

Explanation:

Step1: Calculate \( m\angle B \)

In the triangle, the exterior angle at the base (48°) is equal to the sum of the two non - adjacent interior angles (33° and \( m\angle B \)). So we have the equation \( 48^{\circ}=33^{\circ}+m\angle B \). Solving for \( m\angle B \), we get \( m\angle B = 48^{\circ}-33^{\circ}=15^{\circ} \).

Step2: Calculate \( m\angle A \)

In a triangle, the sum of interior angles is \( 180^{\circ} \). For the triangle with angle of elevation 48°, the right - angle is \( 90^{\circ} \), so \( m\angle A=180^{\circ}-90^{\circ}-48^{\circ}=42^{\circ}
eq57^{\circ} \).

Step3: Calculate \( h \)

Using the Law of Sines in the triangle with sides and angles related to the building height. First, from the triangle with angle 33° and 15°, the side opposite 15° is 20 ft. Let the side opposite 33° be \( c \). By the Law of Sines \( \frac{20}{\sin15^{\circ}}=\frac{c}{\sin33^{\circ}} \), \( c = \frac{20\sin33^{\circ}}{\sin15^{\circ}}\approx\frac{20\times0.5446}{0.2588}\approx41.97 \) ft. Then, in the right - triangle with angle 48°, \( \sin48^{\circ}=\frac{h}{c} \), so \( h = c\sin48^{\circ}\approx41.97\times0.7431\approx31.20
eq8.09 \) ft.

Answer:

The correct measurement is \( \boldsymbol{m\angle B = 15^{\circ}} \) (and we can see that \( m\angle A = 57^{\circ} \) and \( h\approx8.09\mathrm{ft} \) are incorrect. Since we need to choose two correct answers, but from our calculation only \( m\angle B = 15^{\circ} \) is correct among the given options? Wait, maybe there are other options not fully shown. But based on the given options and calculation:

The correct options (assuming there are two correct among the shown and maybe others) from the shown: \( m\angle B = 15^{\circ} \) is correct. If we assume there is another correct one, but from our calculation \( m\angle A = 57^{\circ} \) is wrong and \( h\approx8.09\mathrm{ft} \) is wrong. Maybe there are other options not in the visible part. But for the shown options, the correct one is \( m\angle B = 15^{\circ} \). If we have to choose two, maybe there is a mistake in my calculation or the problem has other parts. But based on the given:

The correct answers (from the shown options and calculation) are:

• \( m\angle B = 15^{\circ} \)

(If there is another correct option, maybe due to incomplete problem, but with the given, this is one correct)