Question

a radio tower has a 27 - foot shadow cast by the sun. if the angle from the tip of the shadow to the top of the tower is 70°, what is the height of the radio tower? round your solution to four decimal places.

Explanation:

Step1: Identify the trigonometric relation

We have a right - triangle where the length of the shadow is the adjacent side ($x = 27$ feet) to the given angle $\theta=70^{\circ}$, and the height of the tower is the opposite side ($y$). We use the tangent function $\tan\theta=\frac{y}{x}$.

Step2: Solve for the height $y$

Since $\tan\theta=\frac{y}{x}$, then $y = x\tan\theta$. Substitute $x = 27$ and $\theta = 70^{\circ}$ into the formula. We know that $\tan70^{\circ}\approx2.7475$, so $y=27\times\tan70^{\circ}$.
$y = 27\times2.7475=74.1825$

Answer:

$74.1825$