I tried to solve this but can't seem to find a procedure to solve this:
To measure the height of Lincoln's caricature on Mt. Rushmore, two sightings 800 feet from the base of the mountain are of Lincoln's face is 32 degrees and the angle of elevation to the top is 35 degrees, what is the height of Lincoln's face?
So far I got tan32= x/800+a and tan35= y/800+a
The horizontal distance H is given.
So the height h1 of the face from the observer's instrument
= H tan(32°)
The height h2 of the top from the observer's instrument
= H tan(35°)
So the height of the face is h2-h1.
because the sightings were 800 from th base of the mountain shouldn't it be 800 + a ?
When surveyors measure distance, they measure the horizontal distance, even if the distance they actually measured is at an angle.
Recall that tangent(θ) of an angle of elevation is opposite(height)/adjacent(horizontal).
So when the question says that the measures were made 800 ft from the base, it means that the horizontal distance is 800 ft. The vertical distance is therefore 800' tan(θ) where θ is the measured angle.
800' (horizontal) cannot be algebraically added to a (vertical), which is the height of the face. The two measurements are not in the same direction, so cannot be added together numerically.
Draw a diagram and see if you see the light. If not, post again.
blah blah blah.
To solve this problem, you can use trigonometry, specifically the tangent function.
First, let's label the unknown height of Lincoln's face as "h", and the distance between the two sightings and the base of the mountain as "d". In this case, d is given as 800 feet.
Now, let's analyze the problem. We have two angles of elevation: 32 degrees and 35 degrees. The angle of elevation is the angle between the line of sight from the observer and the horizontal line.
From the given information, we can create two right triangles. In the first triangle, the opposite side is h (the height of Lincoln's face), and the adjacent side is d (800 feet). Using the tangent function, we can write the equation:
tan(32 degrees) = h / 800
Similarly, in the second triangle, the opposite side is also h (the same height of Lincoln's face), and the adjacent side is also d (800 feet). Using the tangent function again, we can write the equation:
tan(35 degrees) = h / 800
Now, we have a system of two equations with two unknowns (h and d). To solve this system, we can use either substitution or elimination. I will proceed using substitution.
Rearrange the first equation to solve for h:
h = 800 * tan(32 degrees)
Now, substitute this expression for h in the second equation:
tan(35 degrees) = (800 * tan(32 degrees)) / 800
Simplifying:
tan(35 degrees) = tan(32 degrees)
At this point, we can use a scientific calculator or a trigonometric identity table to find the angle that satisfies this equation. In this case, both angles are similar, so we can conclude that the height of Lincoln's face is the same as the distance from the base of the mountain, which is 800 feet.
Therefore, the height of Lincoln's face on Mt. Rushmore is 800 feet.