If you place a 29-foot ladder against the top of a building and the bottom of the ladder is 24 feet from the bottom of the building, How tall is the building? Round to the nearest tenth of a foot. 

If you place a 29-foot ladder against the top of a building and the bottom of the ladder is 24 feet from the bottom of the building, How tall is the building? Round to the nearest tenth of a foot.

The building is approximately 16 feet tall when rounded to the nearest tenth of a foot.

To find the height of the building, you can use the Pythagorean theorem, which states that in a right triangle, the square of the length of the hypotenuse (the longest side, which is the ladder in this case) is equal to the sum of the squares of the lengths of the other two sides (the height of the building and the distance from the building to the ladder's base).

Let's denote:

• The height of the building as "h"
• The distance from the building to the ladder's base as "d"
• The length of the ladder as "L"

Given: d = 24 feet L = 29 feet

Using the Pythagorean theorem:

L^2 = h^2 + d^2

29^2 = h^2 + 24^2

841 = h^2 + 576

Subtracting 576 from both sides:

h^2 = 841 - 576

h^2 = 265

Taking the square root of both sides:

h = sqrt(265)

h ≈ 16.3 feet

So, the building is approximately 16.3 feet tall when rounded to the nearest tenth of a foot.

Applications of Pythagorean Theorem

The Pythagorean Theorem, which states that in a right-angled triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides, has numerous applications across various fields:

1. Geometry: The Pythagorean Theorem is fundamental in geometry for solving problems involving right triangles. It helps in finding missing side lengths or angles in triangles.

2. Architecture and Construction: Architects and builders use the Pythagorean Theorem to ensure that structures are built correctly and to measure distances accurately. For example, it can be used to ensure that walls are perpendicular to each other or to determine the length of diagonal bracing in buildings.

3. Surveying and Mapping: Surveyors use the Pythagorean Theorem to measure distances and create accurate maps. By applying it to right triangles formed between known points, they can calculate distances between those points.

4. Navigation: Navigators use the Pythagorean Theorem to calculate distances between points on maps, especially when determining the shortest distance between two locations, such as in maritime or aerial navigation.

5. Physics: The Pythagorean Theorem is used in physics, particularly in mechanics and kinematics, to analyze motion and forces. For example, it can be used to calculate the resultant of two perpendicular forces.

6. Engineering: Engineers use the Pythagorean Theorem in various applications, such as designing structures, analyzing stresses and strains, and calculating distances in electronic circuits.

7. Computer Graphics: In computer graphics, the Pythagorean Theorem is used extensively to calculate distances, angles, and positions of objects in a two-dimensional or three-dimensional space.

8. Trigonometry: The Pythagorean Theorem is closely related to trigonometric functions and identities. It serves as a foundation for understanding trigonometry and is used in solving trigonometric problems.

9. Optics: In optics, the Pythagorean Theorem is used to calculate distances and angles in optical systems, such as in the design of lenses and mirrors.

10. Finance: The Pythagorean Theorem is sometimes used in financial calculations, such as in the calculation of the length of the diagonal in a bond or the distance between two points on a stock market chart.

These are just a few examples of the many applications of the Pythagorean Theorem in various fields. Its simplicity and versatility make it a valuable tool in problem-solving across different disciplines.