Question

the base of a pyramid is a square with sides 600 ft long, and the height of the pyramid is 200 ft. find the angle of elevation of the edge indicated in the figure to two significant digits. (hint: the base of the triangle in the figure is half the diagonal of the square base of the pyramid.) the angle is given by θ=□°. (round to the nearest degree as needed.)

Explanation:

Step1: Find half the diagonal of the square base

The diagonal of a square with side length \( s \) is \( s\sqrt{2} \). For \( s = 600 \) ft, the diagonal is \( 600\sqrt{2} \) ft. Half of this diagonal is \( \frac{600\sqrt{2}}{2}= 300\sqrt{2}\approx300\times1.4142 = 424.26 \) ft.

Step2: Use tangent to find the angle

We have a right triangle where the opposite side to angle \( \theta \) is the height of the pyramid (200 ft) and the adjacent side is half the diagonal (≈424.26 ft). The tangent of an angle \( \theta \) in a right triangle is \( \tan\theta=\frac{\text{opposite}}{\text{adjacent}} \). So \( \tan\theta=\frac{200}{424.26}\approx0.4714 \).

Step3: Find the angle

To find \( \theta \), we take the arctangent (inverse tangent) of 0.4714. \( \theta=\arctan(0.4714)\approx25.26^\circ \), which rounds to 25° (to two significant digits or nearest degree).

Answer:

\( 25 \)