Question

solve. round answers to the nearest tenth when necessary (one decimal place).
a radio station is building a new transmitter tower. the tower has four guy wires to keep the tower steady. each guy wire is 650 meters long and forms a $74^\circ$ angle of depression with the ground. how tall is the tower?
_______ meters

Explanation:

Step1: Relate angle of depression to triangle angle

The angle of depression from the top of the tower to the ground is equal to the angle of elevation from the ground to the top of the tower, so the angle inside the right triangle (between the ground and the guy wire) is $74^\circ$. The guy wire is the hypotenuse of the right triangle, and the tower height is the opposite side to this angle.

Step2: Use sine function to solve for height

The sine of an angle in a right triangle is $\frac{\text{opposite}}{\text{hypotenuse}}$. Let $h$ = tower height.
$$\sin(74^\circ) = \frac{h}{650}$$
Rearrange to solve for $h$:
$$h = 650 \times \sin(74^\circ)$$

Step3: Calculate the value

Using a calculator, $\sin(74^\circ) \approx 0.9613$
$$h \approx 650 \times 0.9613 = 624.845$$
Round to the nearest tenth: $h \approx 624.8$

Answer:

624.8 meters