To find the height of a square pyramid given the slant height and the base edge, we can use the Pythagorean theorem.
First, we'll denote:
- The height of the tent (pyramid) as \( h \).
- The base edge of the pyramid as \( s = 20 \) meters.
- The slant height as \( l = 13.5 \) meters.
In a square pyramid, the vertical height \( h \), the slant height \( l \), and half the length of the base edge \( \frac{s}{2} \) form a right triangle. The relationship can be expressed as follows:
\[ l^2 = h^2 + \left(\frac{s}{2}\right)^2 \]
First, calculate \( \frac{s}{2} \):
\[ \frac{s}{2} = \frac{20}{2} = 10 \text{ meters} \]
Now substitute \( l \) and \( \frac{s}{2} \) into the Pythagorean theorem equation:
\[ (13.5)^2 = h^2 + (10)^2 \]
Calculating \( (10)^2 \) and \( (13.5)^2 \):
\[ (10)^2 = 100 \] \[ (13.5)^2 = 182.25 \]
Substituting back into the equation:
\[ 182.25 = h^2 + 100 \]
Now, isolate \( h^2 \):
\[ h^2 = 182.25 - 100 = 82.25 \]
Taking the square root to find \( h \):
\[ h = \sqrt{82.25} \approx 9.1 \text{ meters} \]
Thus, the height of the tent is approximately 9.1 meters when rounded to the nearest tenth.
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