Question

use the information in the diagram and the table below to find the length of the wire that supports the tower. diagram: right triangle with 60° angle at the base, base labeled 120 ft, hypotenuse labeled \wire\, vertical side labeled \tower\. table: $sin 30^circ = \frac{1}{2}$, $sin 45^circ = \frac{sqrt{2}}{2}$, $sin 60^circ = \frac{sqrt{3}}{2}$; $cos 30^circ = \frac{sqrt{3}}{2}$, $cos 45^circ = \frac{sqrt{2}}{2}$, $cos 60^circ = \frac{1}{2}$. caption: exact values of common trigonometric functions. options: $120sqrt{2}$ ft, $240$ ft, $120$ ft, $120sqrt{3}$ ft.

Explanation:

Step1: Identify the triangle type

We have a right - triangle with one angle \(60^{\circ}\), the adjacent side to the \(60^{\circ}\) angle is \(120\) ft, and we need to find the length of the wire (the hypotenuse) or maybe the height? Wait, no, the wire is the hypotenuse? Wait, no, let's check the trigonometric ratio. Wait, the angle at the base is \(60^{\circ}\), the adjacent side (base) is \(120\) ft, and we can use the cosine function. Wait, \(\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}}\), but wait, \(\cos60^{\circ}=\frac{1}{2}\), and \(\cos\theta = \frac{\text{adjacent}}{\text{hypotenuse}}\), so \(\cos60^{\circ}=\frac{120}{\text{wire length}}\)? Wait, no, maybe it's a 30 - 60 - 90 triangle. Wait, in a 30 - 60 - 90 triangle, the sides are in the ratio \(1:\sqrt{3}:2\), where the side opposite \(30^{\circ}\) is the shortest one. Wait, if the angle is \(60^{\circ}\), and the adjacent side (let's say the base) is \(120\) ft, and we want to find the hypotenuse (the wire). Wait, \(\cos60^{\circ}=\frac{\text{adjacent}}{\text{hypotenuse}}\), so \(\text{hypotenuse}=\frac{\text{adjacent}}{\cos60^{\circ}}\). Since \(\cos60^{\circ}=\frac{1}{2}\), then \(\text{hypotenuse}=\frac{120}{\frac{1}{2}} = 240\) ft? Wait, no, wait maybe I got the angle wrong. Wait, maybe the angle is \(60^{\circ}\), and the adjacent side is \(120\), but maybe it's the side adjacent to \(60^{\circ}\), and we can also use the sine function for the height. Wait, no, let's re - examine. The triangle is a right - triangle, with angle \(60^{\circ}\) at the base, base length \(120\) ft, and we need to find the length of the wire (the hypotenuse) or the height? Wait, the options include \(240\) ft, \(120\sqrt{3}\) ft, etc. Wait, let's use the cosine of \(60^{\circ}\). \(\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}}\), so \(\cos60^{\circ}=\frac{120}{L}\), where \(L\) is the length of the wire. Since \(\cos60^{\circ}=\frac{1}{2}\), then \(L=\frac{120}{\cos60^{\circ}}=\frac{120}{\frac{1}{2}} = 240\) ft? Wait, no, that seems off. Wait, maybe it's the sine function. Wait, \(\sin60^{\circ}=\frac{\text{opposite}}{\text{hypotenuse}}\), if the opposite side is the height of the tower, and the hypotenuse is the wire. Wait, no, maybe the base is \(120\) ft, angle \(60^{\circ}\), so the hypotenuse (wire) can be found using \(\cos60^{\circ}=\frac{\text{adjacent}}{\text{hypotenuse}}\), so hypotenuse \(=\frac{120}{\cos60^{\circ}} = 240\) ft. Wait, but let's check the 30 - 60 - 90 triangle ratios. In a 30 - 60 - 90 triangle, the sides are \(x\), \(x\sqrt{3}\), and \(2x\), where \(x\) is the side opposite \(30^{\circ}\), \(x\sqrt{3}\) opposite \(60^{\circ}\), and \(2x\) the hypotenuse. If the angle is \(60^{\circ}\), and the adjacent side (which would be opposite \(30^{\circ}\)) is \(120\) ft, then \(x = 120\), so the hypotenuse (wire) is \(2x=240\) ft. Yes, that makes sense.

Step2: Calculate the length of the wire

We know that in a right - triangle, \(\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}}\). Here, \(\theta = 60^{\circ}\), adjacent side \(= 120\) ft, and \(\cos60^{\circ}=\frac{1}{2}\). Let \(L\) be the length of the wire (hypotenuse). Then:
\[
\cos60^{\circ}=\frac{120}{L}
\]
Substitute \(\cos60^{\circ}=\frac{1}{2}\) into the equation:
\[
\frac{1}{2}=\frac{120}{L}
\]
Cross - multiply to get \(L\times1 = 120\times2\), so \(L = 240\) ft.

Answer:

240 ft