Math Problem Statement

The angle of elevation to the top of a Building in New York is found to be 5 degrees from the ground at a distance of 1 mile from the base of the building. Using this information, find the height of the building. Round to two decimal places. Hint: 1 mile = 5280 feet

Solution

We can solve this problem using trigonometry, specifically the tangent function in a right triangle.

Step 1: Identify the Given Information

  • The distance from the observer to the base of the building (adjacent side) = 1 mile = 5280 feet
  • The angle of elevation (θ) =
  • The height of the building (opposite side) = ?

Using the tangent function, we have:

tan(θ)=oppositeadjacent\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}

tan(5)=h5280\tan(5^\circ) = \frac{h}{5280}

Step 2: Solve for hh

Rearrange the equation:

h=5280×tan(5)h = 5280 \times \tan(5^\circ)

Using a calculator:

h=5280×0.08749h = 5280 \times 0.08749

h461.07 feeth \approx 461.07 \text{ feet}

Final Answer:

The height of the building is 461.07 feet.


Do you need further clarification or details?

Related Questions:

  1. What would be the height if the angle of elevation was 10 degrees?
  2. How does the height change if the observer moves closer to the building?
  3. What is the distance needed for an angle of elevation of 15 degrees to the same height?
  4. How would the calculation change if the building was on an inclined plane?
  5. How do we measure the angle of elevation accurately in real-world applications?

Tip:

When using trigonometric functions, always ensure your calculator is in degree mode (not radians) unless specified otherwise.

Ask a new question for Free

By Image

Drop file here or Click Here to upload

Math Problem Analysis

Mathematical Concepts

Trigonometry
Tangent Function
Right Triangles

Formulas

tan(θ) = opposite / adjacent

Theorems

Trigonometric Ratios

Suitable Grade Level

Grades 9-12