Problem 47 - Contemporary Precalculus

Question

A 50 -foot-high flagpole stands on top of a building. From a point on the ground, the angle of elevation of the top of the pole is \(43^{\circ},\) and the angle of elevation of the bottom of the pole is \(40^{\circ} .\) How high is the building?

Step-by-Step Solution

Answer

Answer: The approximate height of the building is 175.66 feet.

Step 1: Draw a diagram and label the variables

To better understand the problem, it's helpful to draw a diagram: 1. Draw a right-angled triangle, with one of the angles being the angle of elevation to the top of the flagpole (\(43^{\circ}\)). This represents one triangle formed by the ground, the observation point, and the top of the flagpole. 2. Draw a second right-angled triangle, with one of the angles being the angle of elevation to the bottom of the flagpole (\(40^{\circ}\)). This represents another triangle formed by the ground, the observation point, and the bottom of the flagpole. 3. Label the distance from the base of the building to the observation point as \(x\). 4. Label the height of the building as \(y\). 5. Label the height of the flagpole as \(f=50\) feet. Note that the two triangles share a common side \(x\), which will be useful for our calculation.

Step 2: Setting up the equations using trigonometry

Let's set up two trigonometric equations for the two triangles: 1. For the angle of elevation to the top of the flagpole (\(43^{\circ}\)): \(\tan(43^{\circ}) = \frac{y + f}{x}\) 2. For the angle of elevation to the bottom of the flagpole (\(40^{\circ}\)): \(\tan(40^{\circ}) = \frac{y}{x}\) We'll solve these equations to find the height of the building (\(y\)).

Step 3: Solve for x

First, let's solve the second equation for \(x\): \(x = \frac{y}{\tan(40^{\circ})}\)

Step 4: Substitute x into the first equation

Now, we'll substitute the expression for \(x\) from step 3 into the first equation to have one equation with only one variable, \(y\): \(\tan(43^{\circ}) = \frac{y + f}{\frac{y}{\tan(40^{\circ})}}\)

Step 5: Solve for y

Simplify and solve the equation for \(y\): \((y \tan(43^{\circ})) \tan(40^{\circ}) = y + f\) \(y\left(\tan(43^{\circ}) \tan(40^{\circ}) - 1\right) = f\) \(y = \frac{f}{\tan(43^{\circ}) \tan(40^{\circ}) - 1}\)

Step 6: Plug in f and calculate y

Substitute \(f=50\) feet into the equation and calculate the height of the building \(y\): \(y = \frac{50}{\tan(43^{\circ}) \tan(40^{\circ}) - 1} \approx 175.66\) feet So, the height of the building is approximately \(175.66\) feet.

Key Concepts

Right-Angled Triangles

Right-angled triangles are an essential component of this problem because they allow us to use trigonometric functions like the tangent function to solve for unknown lengths. A right-angled triangle has one angle that is exactly 90 degrees. This forms three sides: the hypotenuse, opposite side, and adjacent side.

• The hypotenuse is the longest side and is opposite the right angle.
• The opposite side is opposite the angle of interest (in this case, the angle of elevation).
• The adjacent side is the one that forms the angle of interest along with the hypotenuse.

In problems involving right-angled triangles like this one, it's crucial to set up correctly labeled diagrams. These diagrams help visualize the relationships between angles and the sides, making it easier to apply trigonometric ratios.

Angle of Elevation

The angle of elevation is a critical concept when solving problems involving heights and distances, as it looks upward from the observation point to the object. In simpler terms, it's the angle formed by the horizontal line from the observer's eye to the line of sight upwards to the object. These angles help us relate heights to horizontal distances.
For solving the given problem, two angles of elevation are vital:

• The angle to the top of the flagpole, which is given as \(43^{\circ}\).
• The angle to the bottom of the flagpole, given as \(40^{\circ}\).

Measuring these angles allows us to set up equations that can ultimately help determine the height of the building using trigonometry.

Tangent Function

The tangent function is one of the primary trigonometric functions used to connect an angle of a right-angled triangle to the lengths of the opposite and adjacent sides. The tangent of an angle is defined as the ratio of the length of the opposite side to the length of the adjacent side: \(\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}\).
This function is especially useful for calculating unknown lengths in a triangle when one angle and one side length are known. In this exercise, we can use the tangent function to set up the following equations:

• For the angle at the top of the pole: \(\tan(43^{\circ}) = \frac{y + f}{x}\).
• For the angle at the bottom of the pole: \(\tan(40^{\circ}) = \frac{y}{x}\).

These equations significantly aid in finding the height of the building once we substitute appropriate values and solve.

Solving Equations

Solving equations is a systematic way to find unknown values given certain conditions and relationships. In this context, solving for the unknown height of the building involves a series of steps using algebraic manipulation.
To begin, we first solve one of the equations to express one variable in terms of the other. For example, using \(x = \frac{y}{\tan(40^{\circ})}\), we express x in terms of y and insert it back into another equation.
Substituting \(x\) allows us to convert the problem into one with just a single variable, making it easier to solve. The trick is in the delicate balance of isolating and substituting variables to reduce complexity and reach the final answer.
In cases like this problem, simplifying the equation to the form \(y = \frac{f}{\tan(43^{\circ}) \tan(40^{\circ}) - 1}\) involves rearranging and factoring cleverly. Performing these steps accurately yields the building's height.