Question

an electrician leans an extension ladder against the outside wall of a house so that it reaches an electric box 31 feet up. the ladder makes an angle of 78° with the ground. find the length of the ladder. round your answer to the nearest tenth of a foot if necessary.

Explanation:

Step1: Identify the trigonometric relationship

We have a right triangle where the height (opposite side to the angle) is 31 feet, the angle with the ground is \(78^\circ\), and the ladder is the hypotenuse (\(h\)) we need to find. We use the sine function: \(\sin(\theta)=\frac{\text{opposite}}{\text{hypotenuse}}\).

Step2: Set up the equation

\(\sin(78^\circ)=\frac{31}{h}\)

Step3: Solve for \(h\)

Rearrange the equation: \(h = \frac{31}{\sin(78^\circ)}\)

Step4: Calculate the value

First, find \(\sin(78^\circ)\approx0.9781\). Then \(h=\frac{31}{0.9781}\approx31.7\)

Answer:

The length of the ladder is approximately 31.7 feet.