Problem 52 - Algebra and Trigonometry

Question

A pine tree growing on a hillside makes a \(75^{\circ}\) angle with the hill. From a point 80 feet up the hill, the angle of elevation to the top of the tree is \(62^{\circ}\) and the angle of depression to the bottom is \(23^{\circ} .\) Find, to the nearest tenth of a foot, the height of the tree.

Step-by-Step Solution

Answer

The height of the tree is calculated to be \(h\) feet, rounded to the nearest tenth of a foot. The appropriate mathematical calculations and formulas applied to the provided angles and distances would give the specific value for \(h\).

Step 1: Determine the triangle's dimensions

Firstly, the problem can be decomposed into two right triangles: one triangle from the point on the hill to the bottom of the tree and a second triangle from the point on the hill to the top of the tree. The hill can be considered as the base in both cases.

Step 2: Calculate distance to the bottom of the tree

For the first triangle, it has a side on the hill of length 80 feet and its corresponding adjacent angle of \(23^{\circ}\). The distance (d1) from the point on the hill to the bottom of the tree can be calculated using the tangent of the angle, defined as opposite over adjacent. Hence, we have \(d1 = 80 * tan(23^{\circ})\)

Step 3: Calculate distance to the top of the tree

For the second triangle, it has the same side on the hill of length 80 feet and a corresponding adjacent angle of \(62^{\circ}\). The distance (d2) from the point on the hill to the top of the tree can be calculated in the same way. Hence, we have \(d2 = 80 * tan(62^{\circ})\)

Step 4: Calculate the height of the tree

The height of the tree can be found by taking the difference of the two distances found in step 2 and 3. The tree's height is angled on the hill, therefore we use cosine of \(75^{\circ}\) to represent this. Hence, the height (h) of the tree is \(h = (d2 - d1) / cos(75^{\circ})\)

Key Concepts

Trigonometry

Trigonometry is an exciting branch of mathematics that deals with the relationships between the sides and angles of triangles, especially right-angled triangles. It is essential for solving problems involving angles and distances, prevalent in various fields such as astronomy, engineering, and construction. In the scenario of the pine tree on a hill, we apply trigonometric principles to deduce the height of the tree based on given angles of elevation and depression.

With trigonometry, we use trigonometric ratios—sine, cosine, and tangent—to relate the angles of a triangle to its side lengths. These fundamental ratios enable us to solve for unknown sides or angles when certain other dimensions are known. The pine tree problem requires an understanding of tangent ratios to find the distances from the observer's point on the hill to the top and bottom of the tree, as well as the use of cosine to adjust for the hill's angle.

Right Triangle Trigonometry

Right triangle trigonometry is specifically concerned with the properties of right-angled triangles, where one angle is always 90 degrees. The sides of a right triangle are referred to as the opposite, adjacent, and hypotenuse, with the hypotenuse being the side opposite the right angle and the longest side of the triangle. In our tree problem, we are effectively dealing with two separate right triangles: one from the point up the hill to the bottom of the tree and another from the same point to the top of the tree.

In each case, the side along the hill's surface is considered the adjacent side relative to the given angles of elevation and depression. By using the definitions of the trigonometric ratios, specifically the tangent, we are able to find the length of the sides opposite the given angles, thereby determining the distance to the top and bottom of the tree.

Tangent of an Angle

The tangent of an angle in a right triangle is a ratio that represents the relationship between the length of the side opposite to the angle and the length of the side adjacent to the angle. It is one of the key trigonometric functions and is particularly useful for solving problems where we need to find heights or distances that are not directly measurable.

In the context of the pine tree exercise, the angles of elevation and depression provide us with two angles from which we can calculate the distances using the tangent function. Given by the formula \(\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}\), where \(\theta\) is the angle, we use the angles of 62 and 23 degrees to find the respective distances to the top and bottom of the tree, with 80 feet being the adjacent side length for both angles.

Calculating Distances Using Trigonometry

Trigonometry makes calculating distances that are not straightforwardly measurable quite efficient. When we can't measure a distance directly—such as the height of a tree on an inclined hill—we can use trigonometry to find missing lengths. In simple terms, we create an imaginary right triangle (or triangles in this case) and then use known values such as angles and side lengths to determine the unknowns through trigonometric ratios.

The calculation of the tree's height is done in steps using the angles of elevation and depression and the trigonometric function tangent to gauge distances 'd1' and 'd2.' We then determine the difference between these distances and adjust them for the incline of the hill using the cosine function. This multistep approach is a demonstration of applying trigonometry to deduce real-world distances.