Question

a surveyor found the angle of elevation from the ground to the top of the building at two locations 20 feet apart as shown below
which measurements are correct? round side lengths to the nearest hundredth. check all that apply.
$m\angle a = 57^\circ$
$m\angle b = 15^\circ$
$c \approx 10.89$ ft
$h \approx 8.09$ ft
$h \approx 31.28$ ft

Explanation:

Step1: Calculate $\angle A$

The exterior angle of a triangle equals the sum of remote interior angles.
$\angle A = 48^\circ - 33^\circ = 15^\circ$

Step2: Calculate $\angle B$

Sum of angles in a triangle is $180^\circ$.
$\angle B = 180^\circ - 15^\circ - 33^\circ = 132^\circ$

Step3: Find side $c$ via Law of Sines

$\frac{c}{\sin 33^\circ} = \frac{20}{\sin 132^\circ}$
$c = \frac{20 \times \sin 33^\circ}{\sin 132^\circ} \approx \frac{20 \times 0.5446}{0.7431} \approx 14.69$ (corrected calculation)

Step4: Find height $h$

Use right triangle with $48^\circ$ angle. Let base be $x$, so $h = x \tan 48^\circ$. From the other triangle: $h = (x+20) \tan 33^\circ$. Set equal:
$x \tan 48^\circ = (x+20) \tan 33^\circ$
$x (\tan 48^\circ - \tan 33^\circ) = 20 \tan 33^\circ$
$x = \frac{20 \times 0.6494}{1.1106 - 0.6494} \approx \frac{12.988}{0.4612} \approx 28.16$
$h = 28.16 \times \tan 48^\circ \approx 28.16 \times 1.1106 \approx 31.28$ ft

Step5: Verify correct options

• $m\angle A = 15^\circ$ (not $57^\circ$)
• $m\angle B = 132^\circ$ (not $15^\circ$)
• $c \approx 14.69$ ft (not $10.89$ ft)
• $h \approx 31.28$ ft (correct)

Answer:

$\boldsymbol{h \approx 31.28\ \text{ft}}$