Problem 52 - Trigonometry

Question

Gina is standing near a building and notices that the angle of elevation to the top of the building is \(68^{\circ}\). She then walks 72 feet further away from the building and notices that the angle of elevation to the top of the building is now only \(51^{\circ}\). Find the height of the building. a. \(25 \mathrm{ft}\) b. \(56 \mathrm{ft}\) c. \(149 \mathrm{ft}\) d. \(180 \mathrm{ft}\)

Step-by-Step Solution

Answer

The height of the building is approximately 180 ft (choice d).

Step 1: Diagram and Variables

Sketch a right triangle representing the building, where the height of the building is the opposite side and the distance from Gina to the building is the adjacent side for both scenarios. Let the initial distance from the building to where Gina is standing be \(x\) feet and let \( h \) be the height of the building.

Step 2: Set Up Equations Using Right Triangle Trigonometry

For the first position, apply the tangent function: \[ \tan(68^{\circ}) = \frac{h}{x} \] This can be rearranged to: \[ h = x \cdot \tan(68^{\circ}) \] For the second position, after Gina walks 72 feet: \[ \tan(51^{\circ}) = \frac{h}{x + 72} \] This can be rearranged to: \[ h = (x + 72) \cdot \tan(51^{\circ}) \]

Step 3: Set the Equations Equal

Since both expressions are equal to \( h \), we can set them equal to each other:\[ x \cdot \tan(68^{\circ}) = (x + 72) \cdot \tan(51^{\circ}) \]

Step 4: Solve for x

Rearrange and solve for \( x \):\[ x \cdot \tan(68^{\circ}) = x \cdot \tan(51^{\circ}) + 72 \cdot \tan(51^{\circ}) \]\[ x \cdot (\tan(68^{\circ}) - \tan(51^{\circ})) = 72 \cdot \tan(51^{\circ}) \]Divide by \((\tan(68^{\circ}) - \tan(51^{\circ}))\):\[ x = \frac{72 \cdot \tan(51^{\circ})}{\tan(68^{\circ}) - \tan(51^{\circ})} \]

Step 5: Calculate the Value of x

Calculate the tangent values and determine \( x \):\( \tan(68^{\circ}) \approx 2.4751 \) and \( \tan(51^{\circ}) \approx 1.2349 \)Plug these into the equation:\[ x \approx \frac{72 \cdot 1.2349}{2.4751 - 1.2349} \]\[ x \approx \frac{88.9128}{1.2402} \approx 71.69 \] feet

Step 6: Determine the Building's Height

Substitute \( x \approx 71.69 \) feet back into the equation for height:\[ h = x \cdot \tan(68^{\circ}) \]\[ h \approx 71.69 \cdot 2.4751 \approx 177.4 \] feet

Step 7: Choose the Closest Answer

The calculated height is approximately 177.4 feet, so the closest answer choice is:\( 180 \) ft.

Key Concepts

Angle of Elevation

The angle of elevation is a concept commonly used in trigonometry. It is the angle formed between the line of sight from an observer to an object and the horizontal plane of the observer.
This concept is crucial when dealing with problems involving height and distance, such as estimating the height of a building or a tree.

When you stand on the ground looking up at the top of a building, the angle your eyes make with the horizontal line of sight to the building's top is what we call the angle of elevation. This angle can help you find the building's height using trigonometric functions like tangent.

• The angle of elevation increases as you get closer to the object.
• The angle decreases as you move further away.

Observing the change in angle as you move can help calculate not only the height but also the original distance from the object.

Right Triangle Trigonometry

Right triangle trigonometry involves analyzing the relationships between the angles and sides of a right-angled triangle. A right-angled triangle has one angle measuring 90 degrees, with the other two angles being acute.

In these types of triangles, certain functions—sine, cosine, and tangent—are used to relate these angles to the side lengths. These relationships are formulaic expressions:

• Sine, \((\sin(\theta))\), is the ratio of the opposite side to the hypotenuse.
• Cosine, \( (\cos(\theta))\), is the ratio of the adjacent side to the hypotenuse.
• Tangent, \( (\tan(\theta)) \), is the ratio of the opposite side to the adjacent side.

These functions allow us to solve for missing sides or angles when given limited information. In the given problem, right triangle trigonometry helps us create equations that express the height of the building using the tangent of the angles of elevation.

Tangent Function

The tangent function is particularly useful in right triangle trigonometry, especially when dealing with problems involving the angle of elevation. It relates the length of the opposite side to the length of the adjacent side. This makes it useful for finding heights, as in our exercise.

Recall that the tangent of an angle \( (\theta) \) in a right triangle is given by the formula: \( an(\theta) = \frac{\text{opposite}}{\text{adjacent}}\). For our exercise:

• In the first scenario, we relate the height of the building (opposite side) to the initial distance (adjacent side) using \( \tan(68^{\circ}) \).
• In the second scenario, the opposite side remains the height of the building, but the adjacent side becomes greater due to the additional 72 feet.

The equations formed using these relationships allow us to solve for unknown variables, turning a complex problem into a system of solvable algebraic expressions. By relating both equations for the building height to the two angles observed, we can find the building's height with mathematical precision.