Problem 2 - Technical Mathematics with Calculus

Question

From a point on level ground, the angles of elevation of the top and the bottom of an antenna standing on top of a building are \(32.6^{\circ}\) and \(27.8^{\circ},\) respectively. If the building is \(125 \mathrm{ft}\) high, how tall is the antenna? Remember that angles of elevation or depression are always measured from the horizontal.

Step-by-Step Solution

Answer

The height of the antenna is approximately 260.60 feet.

Step 1: Understand the problem

The problem describes a scenario where an observer is looking at the top and bottom of an antenna from the same point on level ground. There are two angles of elevation mentioned; one to the bottom of the antenna (which effectively is the top of the building since the antenna stands on it), and one to the top of the antenna. The goal is to find the height of the antenna. As the observer's line of sight to the bottom and top of the antenna creates two right-angled triangles with the building, we can use trigonometric functions to solve for the antenna's height.

Step 2: Assign variables to the unknown

Let the height of the antenna be represented by the variable 'h'. We are given the building's height as 125 ft. We can now set up our equations using the angles of elevation given.

Step 3: Write the equation for the angle to the top of the antenna

Using the tangent of the angle of elevation to the top of the antenna, we have the equation \[ \tan(32.6^\circ) = \frac{{125 + h}}{{d}} \] where 'd' is the horizontal distance from the observation point to the building, and 'h' is the height of the antenna we are looking for.

Step 4: Write the equation for the angle to the top of the building

Similarly, using the tangent of the angle of elevation to the top of the building (bottom of the antenna), we have \[ \tan(27.8^\circ) = \frac{{125}}{{d}} \]

Step 5: Solve for distance 'd'

From the equation in Step 4, we can isolate 'd' to find \[ d = \frac{{125}}{{\tan(27.8^\circ)}} \]

Step 6: Insert 'd' into the equation for the angle to the top of the antenna

Substituting the value of 'd' from Step 5 into the equation in Step 3, we get \[ \tan(32.6^\circ) = \frac{{125 + h}}{{\frac{{125}}{{\tan(27.8^\circ)}}}} \]

Step 7: Solve for 'h'

Now, we can solve for 'h' \[ \tan(32.6^\circ) = \frac{{125 + h}}{{\frac{{125}}{{\tan(27.8^\circ)}}}} \] \[ (125 + h) = \tan(32.6^\circ) \times \frac{{125}}{{\tan(27.8^\circ)}} \] \[ h = \left[\tan(32.6^\circ) \times \frac{{125}}{{\tan(27.8^\circ)}} \right] - 125 \]

Step 8: Calculate the height 'h'

By plugging in the values of the tangent functions and doing the arithmetic, we find the value of 'h': \[ h = \left[\tan(32.6^\circ) \times \frac{{125}}{{\tan(27.8^\circ)}} \right] - 125 \] \[ h \approx \left[1.6405 \times \frac{{125}}{{0.5317}} \right] - 125 \] \[ h \approx \left[1.6405 \times 235.0139 \right] - 125 \] \[ h \approx 385.60 \text{ ft} - 125 \text{ ft} \] \[ h \approx 260.60 \text{ ft} \]

Key Concepts

Trigonometric Functions

Trigonometric functions are mathematical tools that relate the angles of a triangle to its sides. They are especially useful in right-angled triangles, where one angle is exactly 90 degrees. The primary trigonometric functions are sine, cosine, and tangent. Each function is a ratio that compares two sides of a right-angled triangle.

In the context of the angles of elevation problem from the exercise, we use the tangent function, which compares the opposite side to the adjacent side of the angle in question. If we know the angle and one side, we can use the tangent function to find the length of the other side. This is exactly what was done in the problem solution to determine the height of the antenna.

Right-angled Triangles

A right-angled triangle is a triangle where one of the angles measures exactly 90 degrees. This property is crucial because it allows the use of trigonometric functions to find missing angles and lengths within the triangle. In right-angled triangles, the side opposite the right angle is called the hypotenuse, and the other two sides are referred to as the adjacent and opposite sides, based on their position in relation to the angle of interest.

When solving problems involving right-angled triangles, such as finding the height of an antenna given the angle of elevation, we utilize the relationship between the sides and angles. The clarity of these mathematical relationships is what makes right-angled triangles a fundamental element in trigonometry and practical applications like surveying, construction, and navigation.

Tangent of an Angle

The tangent of an angle in a right-angled triangle is a trigonometric function that represents the ratio of the length of the side opposite to the angle (the opposite side) to the length of the side adjacent to the angle (the adjacent side). It is commonly abbreviated as 'tan.' This function is extremely useful when dealing with angles of elevation or depression as it directly relates an angle to the proportion of two sides of a triangle.

When using the tangent function in calculating the height of the antenna from the original problem, we set up an equation that incorporates the tangent of the known angles and the height of the building, which allows us to solve for the unknown height of the antenna. The function's simple ratio makes it a straightforward tool for such calculations, provided you remember the correct setup: \[ \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} \. \] For example, in the solution provided, the tangent of the angle of elevation helped to create an equation that could be manipulated to solve for the height of the antenna standing on top of the building.