To find the total height of the building composed of a square prism and a square pyramid, we first need to determine the height of the pyramid using the Pythagorean Theorem.
Given that:
- The base of the square prism is a square with sides of 40 m.
- The height of the square prism is 20 m.
- The slant height of the square pyramid is 24 m.
First, let's find the height (h) of the square pyramid. The slant height (l) of the pyramid and the height of the pyramid form a right triangle with half the base of the pyramid.
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Calculate the half base of the pyramid: \[ \text{Half the base} = \frac{40 \text{ m}}{2} = 20 \text{ m} \]
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Apply the Pythagorean Theorem: In the right triangle formed by the height of the pyramid, the half base of the pyramid, and the slant height, we have: \[ l^2 = h^2 + \left(\frac{\text{base}}{2}\right)^2 \] Substituting the known values: \[ 24^2 = h^2 + 20^2 \] \[ 576 = h^2 + 400 \] \[ h^2 = 576 - 400 \] \[ h^2 = 176 \] \[ h = \sqrt{176} \] Calculating \( h \): \[ h \approx 13.3 \text{ m} \quad (\text{rounded to the nearest tenth}) \]
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Calculate the total height of the building: The total height of the building is the height of the prism plus the height of the pyramid: \[ \text{Total height} = \text{Height of prism} + \text{Height of pyramid} \] \[ \text{Total height} = 20 \text{ m} + 13.3 \text{ m} = 33.3 \text{ m} \]
Therefore, the height of the building is approximately \( \boxed{33.3} \) m.
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