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.
$m\angle b = 15^\circ$
$m\angle a = 57^\circ$
$h \approx 8.09\text{ft}$

Explanation:

Step1: Calculate \( m\angle B \)

In triangle \( ABC \), the sum of angles in a triangle is \( 180^\circ \). The two given angles at the base are \( 33^\circ \) and \( 48^\circ \), but \( \angle B \) is the difference between them. So \( m\angle B = 48^\circ - 33^\circ = 15^\circ \).

Step2: Calculate \( m\angle A \)

\( \angle A \) is supplementary to the \( 48^\circ \) angle? No, wait, in the triangle with height \( h \), the right - triangle: Wait, no, first, in triangle \( ABC \), we can also calculate \( \angle A \) as \( 180^\circ-(90^\circ + 33^\circ) \)? No, let's re - examine. Wait, the angle at the base for the right - triangle (with height \( h \)): the angle adjacent to the side with length related to \( h \) is \( 48^\circ \), so the angle \( \angle A \) in the non - right triangle: Wait, actually, to find \( m\angle A \), we know that in the triangle where we have the two angles at the ground (33° and 48°) and the side of 20 ft. Wait, \( \angle A \) is in the triangle with angles: let's see, the angle at the top of the triangle (the one with the two lines to the building) - the sum of angles in a triangle is \( 180^\circ \). Wait, maybe a better way: For the angle \( \angle A \), in the triangle, we can consider that the angle inside the triangle (the one at point \( A \)) is \( 180 - 48=132^\circ \)? No, that's not right. Wait, let's start over.

First, for \( m\angle B \): The two angles of elevation are \( 33^\circ \) and \( 48^\circ \), so the angle between the two lines of sight ( \( \angle B \)) is \( 48^\circ-33^\circ = 15^\circ \), so \( m\angle B = 15^\circ \) is correct.

For \( m\angle A \): In triangle \( ABC \), we know that one angle is \( 15^\circ \) ( \( \angle B \) ), and the angle at the far - left is \( 33^\circ \), so the third angle ( \( \angle A \)) is \( 180^\circ-(15^\circ + 33^\circ)=132^\circ \)? No, that's not matching the option. Wait, maybe I made a mistake. Wait, the right - triangle: the height \( h \) is opposite to the \( 48^\circ \) angle in one right - triangle and opposite to the \( 33^\circ \) angle in the other? No, the two locations are 20 ft apart. Let's use the Law of Sines. Let's denote the distance from the closer point to the building as \( x \), then the distance from the farther point is \( x + 20 \). In the right - triangle with angle \( 48^\circ \), \( \tan(48^\circ)=\frac{h}{x} \), so \( x=\frac{h}{\tan(48^\circ)} \). In the right - triangle with angle \( 33^\circ \), \( \tan(33^\circ)=\frac{h}{x + 20} \), so \( x+20=\frac{h}{\tan(33^\circ)} \). Substituting \( x \) from the first equation into the second: \( \frac{h}{\tan(48^\circ)}+20=\frac{h}{\tan(33^\circ)} \).

\( h(\frac{1}{\tan(33^\circ)}-\frac{1}{\tan(48^\circ)}) = 20 \)

\( \tan(33^\circ)\approx0.6494 \), \( \tan(48^\circ)\approx1.1106 \)

\( \frac{1}{\tan(33^\circ)}\approx1.5399 \), \( \frac{1}{\tan(48^\circ)}\approx0.9004 \)

\( h(1.5399 - 0.9004)=20 \)

\( h(0.6395)=20 \)

\( h=\frac{20}{0.6395}\approx31.27 \), so \( h\approx8.09 \) is wrong.

Wait, maybe the triangle is the one with sides: the side between the two locations is 20 ft, and the two angles at the ground are \( 33^\circ \) and \( 48^\circ \), so the angle at the top ( \( \angle B \)) is \( 48 - 33 = 15^\circ \), and then using Law of Sines: \( \frac{20}{\sin(15^\circ)}=\frac{c}{\sin(33^\circ)} \), where \( c \) is the distance from the closer point to the building. Then \( c=\frac{20\sin(33^\circ)}{\sin(15^\circ)} \). \( \sin(15^\circ)\approx0.2588 \), \( \sin(33^\circ)\approx0.5446 \)

\( c=\frac{20\times0.5446}{0.2588}\approx41.93 \)

Then \( h = c\sin(48^\circ)\approx41.93\times0.7431\approx31.27 \), so \( h\approx8.09 \) is wrong.

Now, for \( m\angle A \): Wait, maybe \( \angle A \) is \( 180 - 90 - 33=57^\circ \)? Wait, in the right - triangle with angle \( 33^\circ \), the other non - right angle is \( 90 - 33 = 57^\circ \), so \( m\angle A = 57^\circ \) is correct? Wait, maybe I mis - identified the triangle. Let's look at the diagram: point \( A \) is between the two ground points, the building has a right angle with the ground. So the triangle at point \( A \) is a right - triangle with angle \( 48^\circ \) at \( A \), so the other angle ( \( \angle A \) in the option) is \( 90 - 48 = 42^\circ \)? No, this is confusing. Wait, the options are \( m\angle B = 15^\circ \), \( m\angle A = 57^\circ \), \( h\approx8.09 \) ft.

We know that \( 48^\circ-33^\circ = 15^\circ \), so \( m\angle B = 15^\circ \) is correct.

For \( m\angle A \): If we consider the triangle where the angle at the ground is \( 33^\circ \), and it's a right - triangle, then the angle \( \angle A \) (the angle inside the non - right triangle) - wait, maybe the angle \( \angle A \) is \( 180-(90 + 33)=57^\circ \), yes! Because in the right - triangle with angle \( 33^\circ \), the two non - right angles sum to \( 90^\circ \), so if one is \( 33^\circ \), the other is \( 90 - 33 = 57^\circ \), so \( m\angle A = 57^\circ \) is correct.

And \( h\approx8.09 \) is wrong as we calculated \( h\approx31.27 \). So the correct answers are \( m\angle B = 15^\circ \) and \( m\angle A = 57^\circ \).

Answer:

The two correct answers are:

• \( m\angle B = 15^\circ \)
• \( m\angle A = 57^\circ \)