Surveying The distance from point A to the top of the hill...

Surveying The distance from point $A$ to the top of the hill is 2760 ft. The angle of elevation from $A$ to the base of the tower from $A$ the angle of the tower is $28^{\circ}$ and the angle of the tower from $A$ to the top of the tower is $32^{\circ} .$ a. Find the measures of $\angle 1$ and $\angle 2 .$ b. Find to the nearest foot the height of the tower above the top of the hill.

Surveying The distance from point $A$ to the top of the hill is 2760 ft. The angle of elevation from $A$ to the base of the tower from $A$ the angle of the tower is $28^{\circ}$ and the angle of the tower from $A$ to the top of the tower is $32^{\circ} .$
a. Find the measures of $\angle 1$ and $\angle 2 .$
b. Find to the nearest foot the height of the tower above the top of the hill.

Key Concepts

Surveying Techniques

Surveying involves accurately measuring distances and angles to determine positions and elevations of features in the landscape. It relies on a combination of observational measurements and trigonometric calculations to solve real-world problems such as determining the height of structures or the topography of terrain.

Height Calculation Using Angle Differences

When an object is elevated on a hill or has multiple layers of elevation (such as a tower on a hill), the difference between the angles of elevation to different reference points is used to isolate the height of a particular section. This often involves calculating the heights corresponding to each angle using the tangent function and then finding the difference between them.

Tangent Function

The tangent of an angle in a right triangle is the ratio of the opposite side (vertical height) to the adjacent side (horizontal distance). It is a fundamental tool in surveying for computing height differences when the angle of elevation and horizontal distance are known.

Right Triangle Trigonometry

In surveying problems, right triangle trigonometry is used to relate the angles and sides of a right triangle. By setting up triangles with a known horizontal distance, trigonometric ratios like the tangent can be applied to calculate vertical heights from measured angles.

Angle of Elevation

This is the angle between the horizontal line of sight and the line of sight upward to an object. In practical applications like surveying, it is used to determine the height of objects when the distance between the observer and the object is known, using trigonometric relationships.

Transcript

00:01 A of our problem, we can begin by finding the measure of angle one by essentially just subtracting the two angles that we're given.

00:09 So our first angle is going to be right here, and it's 28 degrees.

00:14 And our second angle encompasses both of our lines, which is 32 degrees.

00:20 So to find the angle, the portion of the angle that is not included in this bottom portion, we can see that essentially we're just going to subtract the whole, essentially the entire angle, minus the small portion of our angle.

00:37 So it's going to be equal to 32 degrees minus 28 degrees, which we can see is equal to 4 degrees.

00:46 And so we can go ahead and say that now we know what angle 1 is.

00:51 And so to use law of signs to find angle 2, we would need another piece of information.

00:58 We would essentially either need to know the length of side one or we would need to know this angle and this side.

01:09 And so to find my third angle or angle two, i'm essentially going to be able to use the right triangle, the lower right triangle, which i've drawn right here in order to solve for this angle right here.

01:22 So we know that because it is making a right angle with the ground, this angle up here is equal to 180 degrees minus the 90 of my right angle minus the 28 degrees of our angle that we're given in the problem.

01:41 And so this is equal to 62 degrees.

01:45 And so because we know from this first diagram on the left here that this angle plus.

01:54 This upper angle here, which i'm just going to go ahead and label angle 3, are equal to 180 degrees.

02:00 Because they form a straight line, we can find angle 3 by subtracting 180 minus this angle down here, which we just said was 62 degrees...

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