Question

to measure a stone face carved on the side of a mountain, two sightings 650 feet from the base of the mountain are taken. if the angle of elevation to the bottom of the face is 32° and the angle of elevation to the top is 35°, what is the height of the stone face? the height of the stone face is \boxed{} feet.

Explanation:

Step1: Define variables for heights

Let \( h_1 \) be the height from the base to the bottom of the stone face, and \( h_2 \) be the height from the base to the top of the stone face. The distance from the sighting point to the base is \( d = 650 \) feet. The angle of elevation to the bottom is \( \theta_1 = 32^\circ \) and to the top is \( \theta_2 = 35^\circ \).

Step2: Use tangent function for \( h_1 \)

We know that \( \tan\theta_1=\frac{h_1}{d} \), so \( h_1 = d\times\tan\theta_1 \). Substituting \( d = 650 \) and \( \theta_1 = 32^\circ \), we get \( h_1=650\times\tan(32^\circ) \). Calculating \( \tan(32^\circ)\approx0.6249 \), so \( h_1\approx650\times0.6249 = 406.185 \) feet.

Step3: Use tangent function for \( h_2 \)

Similarly, \( \tan\theta_2=\frac{h_2}{d} \), so \( h_2 = d\times\tan\theta_2 \). Substituting \( d = 650 \) and \( \theta_2 = 35^\circ \), we get \( h_2 = 650\times\tan(35^\circ) \). Calculating \( \tan(35^\circ)\approx0.7002 \), so \( h_2\approx650\times0.7002=455.13 \) feet.

Step4: Calculate the height of the stone face

The height \( h \) of the stone face is \( h = h_2 - h_1 \). Substituting the values of \( h_1 \) and \( h_2 \), we get \( h=455.13 - 406.185 = 48.945\approx49 \) feet (or more precise calculation: \( 650\times(\tan(35^\circ)-\tan(32^\circ)) \)). Let's calculate \( \tan(35^\circ)-\tan(32^\circ)\approx0.7002 - 0.6249 = 0.0753 \). Then \( 650\times0.0753 = 48.945\approx49 \) feet.

Answer:

The height of the stone face is approximately \(\boldsymbol{49}\) feet (or more precisely, around 48.95 feet).