A 40-foot wire stretches from the top of a building to a point on the ground 20 20 feet from the base of the building. If the ground is level and forms a right angle with the building, what is the height, in feet, of the building? Round your answer to the nearest foot.
Question
A 40-foot wire stretches from the top of a building to a point on the ground 20 20 feet from the base of the building. If the ground is level and forms a right angle with the building, what is the height, in feet, of the building? Round your answer to the nearest foot.
Solution
This is a right triangle problem. We can use the Pythagorean theorem to solve it. The Pythagorean theorem states that in a right 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. This can be written as: a² + b² = c².
In this problem, we know:
- The length of the hypotenuse (c) is 40 feet.
- The length of one side (a) is 20 feet.
- We're trying to find the length of the other side (b), which represents the height of the building.
Plugging the known values into the Pythagorean theorem gives us: 20² + b² = 40².
Solving this equation step by step:
- Square the known values: 400 + b² = 1600.
- Subtract 400 from both sides to isolate b²: b² = 1600 - 400.
- Simplify the right side: b² = 1200.
- Take the square root of both sides to solve for b: b = √1200.
The square root of 1200 is approximately 34.64.
So, the height of the building is approximately 35 feet when rounded to the nearest foot.
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