To solve the problem, we can use the Pythagorean theorem, which 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. In this situation, the ladder serves as the hypotenuse, the height of the building is one side, and the distance from the building to the bottom of the ladder is the other side.
The formula can be written as: \[ a^2 + b^2 = c^2 \] where:
- \( c \) is the length of the ladder (hypotenuse),
- \( a \) is the height of the building,
- \( b \) is the distance from the wall to the bottom of the ladder.
Given:
- \( c = 34 \) feet (length of the ladder),
- \( b = 25 \) feet (distance from the ladder to the building).
We want to find the height \( a \) of the building.
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Substitute the known values into the Pythagorean theorem: \[ a^2 + 25^2 = 34^2 \]
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Calculate \( 25^2 \) and \( 34^2 \): \[ 25^2 = 625 \] \[ 34^2 = 1156 \]
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Substitute these values into the equation: \[ a^2 + 625 = 1156 \]
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Solve for \( a^2 \): \[ a^2 = 1156 - 625 \] \[ a^2 = 531 \]
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Finally, we take the square root to find \( a \): \[ a = \sqrt{531} \approx 23.05167380558057 \]
Rounding to the nearest tenth: \[ a \approx 23.1 \]
Therefore, the height of the building is approximately 23.1 feet.