To find the height of the Great Pyramid of Giza, we can use the Pythagorean theorem. The pyramid has a square base, and we can visualize it as having a vertical height, a half-base length, and the slant height forming a right triangle.
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The base edge length is given as \(230.5\) m. Therefore, the half-base length is: \[ \frac{230.5}{2} = 115.25 \text{ m} \]
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We denote the height of the pyramid as \(h\) and the slant height as \(s = 180.2\) m.
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According to the Pythagorean theorem: \[ s^2 = h^2 + \left(\frac{230.5}{2}\right)^2 \]
Plugging in the known values: \[ (180.2)^2 = h^2 + (115.25)^2 \]
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Calculate \((180.2)^2\) and \((115.25)^2\): \[ (180.2)^2 = 32472.04 \] \[ (115.25)^2 = 13276.5625 \]
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Now plug these values into the equation: \[ 32472.04 = h^2 + 13276.5625 \]
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Solve for \(h^2\): \[ h^2 = 32472.04 - 13276.5625 = 19195.4775 \]
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Finally, take the square root to find \(h\): \[ h = \sqrt{19195.4775} \approx 138.578 \]
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Rounding to the nearest tenth: \[ \boxed{138.6} \text{ m} \]
Thus, the height of the pyramid is approximately \(138.6\) meters.
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