Question

the tallest television transmitting tower in the world is in north dakota. from a point on level ground 5280 feet (one mile) from the base of the tower, the angle of elevation is 20.5°. approximate the height of the tower to the nearest foot. 1973 feet (round to the nearest foot.)

Explanation:

Step1: Set up tangent - ratio

We know that in a right - triangle, $\tan\theta=\frac{\text{opposite}}{\text{adjacent}}$. Let the height of the tower be $h$. The adjacent side to the angle of elevation is the distance from the point on the ground to the base of the tower ($d = 5280$ feet) and the angle of elevation $\theta=20.5^{\circ}$. So, $\tan(20.5^{\circ})=\frac{h}{5280}$.

Step2: Solve for $h$

Multiply both sides of the equation by 5280: $h = 5280\times\tan(20.5^{\circ})$.
We know that $\tan(20.5^{\circ})\approx0.3737$. Then $h = 5280\times0.3737= 5280\times\frac{3737}{10000}= \frac{5280\times3737}{10000}=\frac{19731360}{10000}=1973.136$.

Step3: Round the result

Rounding $1973.136$ to the nearest foot gives $h\approx1973$ feet.

Answer:

1973