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 38° and the angle of elevation to the top is 41°, what is the height of the stone face? the height of the stone face is \boxed{} feet. (do not round until the final answer. then round to two decimal places as needed.)

Explanation:

Step1: Define the problem with trigonometry

We have a right triangle situation where the adjacent side (distance from the base) is 650 feet. Let \( h_1 \) be the height to the bottom of the face and \( h_2 \) be the height to the top. The height of the stone face \( h = h_2 - h_1 \). Using the tangent function: \( \tan(\theta)=\frac{\text{opposite}}{\text{adjacent}} \), so \( h_1 = 650\times\tan(38^\circ) \) and \( h_2 = 650\times\tan(41^\circ) \).

Step2: Calculate \( h_1 \) and \( h_2 \)

First, calculate \( h_1 \): \( h_1 = 650\times\tan(38^\circ) \approx 650\times0.7813 = 507.845 \)
Then, calculate \( h_2 \): \( h_2 = 650\times\tan(41^\circ) \approx 650\times0.8693 = 565.045 \)

Step3: Find the height of the stone face

Subtract \( h_1 \) from \( h_2 \): \( h = h_2 - h_1 = 565.045 - 507.845 = 57.2 \) (Wait, no, let's do it more accurately without rounding intermediate steps. Let's use more precise values of tangent. \( \tan(38^\circ)\approx0.7812856265 \), \( \tan(41^\circ)\approx0.8692867378 \)
\( h_1 = 650\times0.7812856265 = 650\times0.7812856265 = 507.8356572 \)
\( h_2 = 650\times0.8692867378 = 565.0363796 \)
\( h = 565.0363796 - 507.8356572 = 57.2007224 \)

Answer:

\( 57.20 \) (rounded to two decimal places)