A building was built to be 10,000 meters tall, but over time the once perfectly vertical building has begun to lean and is now 5 degrees from vertical. What is the height of the bu... A building was built to be 10,000 meters tall, but over time the once perfectly vertical building has begun to lean and is now 5 degrees from vertical. What is the height of the building now? Read more

Steps to Solve

1. Understand the problem setup

The building was originally vertical and has a height of 10,000 meters. It now leans at an angle of 5 degrees from vertical.

2. Draw a right triangle for visualization

When the building leans, it forms a right triangle where:

• The original height (10,000 meters) is the hypotenuse.
• The new height of the building can be represented as the adjacent side of the triangle.
• The angle of interest is the angle between the vertical line and the building (5 degrees).

3. Use the cosine function to find the new height

To find the new height $h$, we can use the cosine of the angle, which is defined as: $$ \cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}} $$ For our situation, this translates to: $$ \cos(5^\circ) = \frac{h}{10,000} $$

4. Rearrange the equation to solve for height

We can rearrange the equation to solve for $h$: $$ h = 10,000 \cdot \cos(5^\circ) $$

5. Calculate the cosine and the new height

Using a calculator: $$ h \approx 10,000 \cdot \cos(5^\circ) \approx 10,000 \cdot 0.9962 \approx 9,962 $$