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 24.5°. approximate the height of the tower to the nearest foot. feet (round to the nearest foot.)

Explanation:

Step1: Set up the tangent - ratio equation

We know that in a right - triangle, $\tan\theta=\frac{\text{opposite}}{\text{adjacent}}$. Let the height of the tower be $h$ (opposite side) and the distance from the base of the tower be $d = 5280$ feet (adjacent side), and the angle of elevation $\theta=24.5^{\circ}$. So, $\tan\theta=\tan(24.5^{\circ})=\frac{h}{5280}$.

Step2: Solve for $h$

Multiply both sides of the equation $\tan(24.5^{\circ})=\frac{h}{5280}$ by 5280. We get $h = 5280\times\tan(24.5^{\circ})$.
We know that $\tan(24.5^{\circ})\approx0.4557$. Then $h = 5280\times0.4557$.
$h=5280\times0.4557 = 2403.096$.

Step3: Round to the nearest foot

Rounding 2403.096 to the nearest foot gives $h\approx2403$ feet.

Answer:

2403