Question

a wire makes a 70° angle with the ground and is attached to the top of a 50 ft antenna. how long is the wire? round to the nearest foot.

Explanation:

Step1: Identify the trig - ratio

We have a right - triangle where the height of the antenna (opposite side to the given angle) is 50 ft and the length of the wire is the hypotenuse $x$. We use the sine function since $\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}$. Here, $\theta = 70^{\circ}$ and the opposite side $a = 50$ ft. So, $\sin70^{\circ}=\frac{50}{x}$.

Step2: Solve for $x$

We can rewrite the equation as $x=\frac{50}{\sin70^{\circ}}$. Since $\sin70^{\circ}\approx0.9397$, then $x=\frac{50}{0.9397}\approx53.2$. Rounding to the nearest foot, $x\approx53$ ft.

Answer:

53 ft