Question

a radio tower has a 99 - foot shadow cast by the sun. if the angle from the tip of the shadow to the top of the tower is 63°, what is the height of the radio tower? round your solution to four decimal places.

Explanation:

Step1: Set up the tangent - ratio

We know that in a right - triangle (where the radio tower is the vertical side, the shadow is the horizontal side, and the line from the tip of the shadow to the top of the tower is the hypotenuse), the tangent of an angle of elevation is given by the ratio of the opposite side to the adjacent side. Let $h$ be the height of the radio tower. The angle of elevation $\theta = 63^{\circ}$ and the adjacent side to the angle $\theta$ is the length of the shadow, $x = 99$ feet. The formula for the tangent of an angle in a right - triangle is $\tan\theta=\frac{opposite}{adjacent}$. So, $\tan(63^{\circ})=\frac{h}{99}$.

Step2: Solve for $h$

We know that $\tan(63^{\circ})\approx1.9626105055$. Then, $h = 99\times\tan(63^{\circ})$. Substitute the value of $\tan(63^{\circ})$ into the equation: $h=99\times1.9626105055$.
$h = 194.2984300445\approx194.2984$.

Answer:

$194.2984$