Recently I took a trip to a big city and I happened to take my clinometer with me. I stood 350 feet away from a building, and found the angle of elevation to be 55°. If my eye height is 5 feet, to the nearest hundredth how tall is the building? 350 ft Type your answer..
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Angles within a circle are feasible to create with the help of different properties of the circle such as radii, tangents, and chords. The radius is the distance from the center of the circle to the circumference of the circle. A tangent is a line made perpendicular to the radius through its endpoint placed on the circle as well as the line drawn at right angles to a tangent across the point of contact when the circle passes through the center of the circle. The chord is a line segment with its endpoints on the circle. A secant line or secant is the infinite extension of the chord.
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A circular arc is the arc of a circle formed by two distinct points. It is a section or segment of the circumference of a circle. A straight line passing through the center connecting the two distinct ends of the arc is termed a semi-circular arc.
![### Mathematical Problem: Calculating the Height of a Building Using Trigonometry
**Problem Statement:**
Recently I took a trip to a big city and I happened to take my clinometer with me. I stood 350 feet away from a building, and found the angle of elevation to be 55°. If my eye height is 5 feet, to the nearest hundredth how tall is the building?
**Diagram Explanation:**
The diagram accompanying this problem presents a right triangle:
1. The horizontal leg (adjacent side) of the triangle represents the distance from the observer to the building, which is 350 feet.
2. The angle of elevation between the observer’s line of sight and the top of the building is 55°.
3. The vertical leg (opposite side) of the triangle represents the height of the building above the observer’s eye height. This is what we need to calculate.
There's an arrow indicating the angle of elevation at 55° from the horizontal line (350 feet) to the top of the building.
**Solution:**
To solve for the height of the building, we use the trigonometric function tangent, which relates the angle of elevation with the opposite and adjacent sides of a right triangle:
\[ \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} \]
Given:
- \(\theta = 55°\)
- adjacent = 350 feet
- eye height = 5 feet
First, calculate the height of the triangle (which is the opposite side) using the tangent function:
\[ \tan(55°) = \frac{\text{opposite}}{350} \]
\[ \text{opposite} = 350 \times \tan(55°) \]
Using a calculator:
\[ \text{opposite} \approx 350 \times 1.4281 \approx 499.83 \text{ feet} \]
Since this is the height from the observer's eye level, we add the observer's eye height to this value to find the total height of the building:
\[ \text{Height of the building} = 499.83 + 5 = 504.83 \text{ feet} \]
**Answer:**
The building is approximately 504.83 feet tall.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fb5b34e65-7f01-4eff-8399-312ff1ddffb7%2Ff3174572-33cc-4a18-bad4-f211709d2bce%2Fw8jch5b_processed.jpeg&w=3840&q=75)
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