From a point on ground level, you measure the angle of...

From a point on ground level, you measure the angle of elevation to the top of a mountain to be $38^{\circ} .$ Then you walk $200 \mathrm{m}$ farther away from the mountain and find that the angle of elevation is now $20^{\circ} .$ Find the height of the mountain. Round the answer to the nearest meter.

Key Concepts

Angle of Elevation

This is the angle between the horizontal line of sight and the line from the observer's eye to the top of an object. It is a fundamental concept in trigonometry and is used extensively in problems involving heights and distances, as it provides a way to relate an object's vertical height to the horizontal distance from the observation point.

Right Triangle Trigonometry

Many real-world problems, especially those involving heights and distances, can be modeled using right triangles. In such triangles, one angle is 90° and the relationships between the sides can be analyzed using trigonometric functions. This approach allows one to relate the height of an object to the distance from its base when the angle of elevation is known.

Tangent Function

The tangent of an angle in a right triangle is the ratio of the length of the opposite side to that of the adjacent side. This function is particularly useful in scenarios where the angle of elevation and the adjacent side (horizontal distance) are known, allowing calculation of the opposite side (height).

Systems of Equations in Trigonometric Contexts

In problems where measurements are taken from multiple positions, multiple trigonometric equations can be formed based on the different angles and distances. These equations share common variables, such as the height of an object, and can be solved simultaneously to determine unknown quantities. This approach integrates both algebra and trigonometry in solving real-world problems.

Rounding and Approximations

Rounding is often used in mathematical problems involving measurement to provide a practical answer that fits within the precision of measurement tools or real-life application. It involves adjusting numerical answers to a specified degree of precision, which is essential when dealing with approximate values derived from trigonometric calculations.

Transcript

00:01 So as we can see, we have this mountain that has height h, and we have two angles of elevation, 1 .38 degrees and 120 degrees, and these two points are 200 meters apart.

00:13 So we have some right triangles here.

00:16 Let's label all our sides, though, or the this side here.

00:20 We'll just call this x for now, so that we can use h and x or x plus 200 to write some trigonometric expressions.

00:28 So we can use the tangent function here.

00:33 Tantion of 38 degrees will equal the opposite side, which is h, over the adjacent side, which is x.

00:43 So we can also write another similar equation with the 20 degrees.

00:47 So tangent of 20 degrees equals h over x plus 200.

00:54 So now we have the system of two equations, and we'll do algebra to solve for h.

01:00 So in this first equation, we can solve for x by multiplying by x and dividing by tangent of 38 degrees on both sides.

01:12 Now we can use this expression for x to substitute into the other equation to help us solve for h.

01:19 So substituting in, we have tangent of 20 degrees equals h over, h over tangent 38 degrees plus 200.

01:30 So we're just inputting this in for x.

01:33 Now to move forward with solving this, let's multiply both sides by this giant denominator here.

01:39 So we have this large denominator.

01:42 We'll multiply both sides by it...

Please give Ace some feedback