To find the height of the Great Pyramid of Giza, we can use the Pythagorean theorem. The pyramid forms a right triangle when we consider the height, half of the base length, and the slant height.
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The edge length of the square base is 230.5 m, so half of the base length is: \[ \text{Half of base length} = \frac{230.5}{2} = 115.25 \text{ m} \]
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The slant height of the pyramid is given as 180.2 m.
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We can represent the height (h) of the pyramid along with the half base and the slant height in a right triangle, where the slant height is the hypotenuse: \[ \text{slant height}^2 = \text{height}^2 + \left(\text{half base length}\right)^2 \] Substituting in the values: \[ 180.2^2 = h^2 + 115.25^2 \]
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Now we calculate \(180.2^2\) and \(115.25^2\): \[ 180.2^2 = 32472.04 \] \[ 115.25^2 = 13274.0625 \]
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Plug these into the equation: \[ 32472.04 = h^2 + 13274.0625 \]
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Now, isolate \(h^2\): \[ h^2 = 32472.04 - 13274.0625 \] \[ h^2 = 19197.9775 \]
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Taking the square root to find \(h\): \[ h = \sqrt{19197.9775} \approx 138.6 \text{ m} \]
Rounding to the nearest tenth, the height of the pyramid is: \[ \boxed{138.6} \text{ m} \]