Question

a wire is attached 35 feet above the ground to a telephone pole. the wire makes an angle of 45° with the ground, as shown below. what is the length of the wire? 35\sqrt{3} feet 65 feet 35 feet 35\sqrt{2} feet

Explanation:

Step1: Identify the trigonometric relationship

We have a right - triangle where the height of the pole (opposite side to the angle) is \(h = 35\) feet and the wire is the hypotenuse \(l\). The angle with the ground \(\theta=45^{\circ}\). Using the sine function \(\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}\), so \(\sin45^{\circ}=\frac{35}{l}\).

Step2: Solve for \(l\)

Since \(\sin45^{\circ}=\frac{\sqrt{2}}{2}\), we have \(\frac{\sqrt{2}}{2}=\frac{35}{l}\). Cross - multiply gives \(l\times\sqrt{2}=35\times2\), then \(l = \frac{70}{\sqrt{2}}\). Rationalize the denominator: \(l=\frac{70\sqrt{2}}{2}=35\sqrt{2}\) feet.

Answer:

\(35\sqrt{2}\text{ feet}\) (the fourth option)