To find the length of the base of the pyramid, we can use the relationship between the height, the angle of elevation, and the length of the base in the right triangle formed.
Given:
- Height (h) of the pyramid = 71 meters
- Angle of elevation (θ) = 42°
We need to find the horizontal distance (d) from the base of the height (71 meters) down to the point directly below the apex of the pyramid, which will help in identifying the base of the pyramid.
Using the tangent function: \[ \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} = \frac{h}{d} \]
Rearranging gives us: \[ d = \frac{h}{\tan(\theta)} \]
Substituting the known values: \[ d = \frac{71}{\tan(42^\circ)} \]
Now we can calculate \(\tan(42^\circ)\): \[ \tan(42^\circ) \approx 0.9004 \]
Now substitute this value into the equation for d: \[ d = \frac{71}{0.9004} \approx 78.8 \text{ meters} \]
The length of the base of a square pyramid is twice this distance (as it represents half the base from the center to one side): \[ \text{Length of the base} = 2d = 2 \times 78.8 \approx 157.6 \text{ meters} \]
Rounding to the nearest whole number gives: \[ \text{Length of the base} \approx 158 \text{ meters} \]
Thus, the length of the base of the pyramid is \(\boxed{158}\) meters.
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