Question

an electrician leans an extension ladder against the outside wall of a house so that it reaches an electri box 31 feet up. the ladder makes an angle of 71° 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 opposite side to the angle \(71^\circ\) is 31 feet (the height on the wall), and the ladder is the hypotenuse \(x\). We use the sine function, which is defined as \(\sin(\theta)=\frac{\text{opposite}}{\text{hypotenuse}}\). So, \(\sin(71^\circ)=\frac{31}{x}\).

Step2: Solve for \(x\)

Rearrange the formula to solve for \(x\): \(x = \frac{31}{\sin(71^\circ)}\). Calculate \(\sin(71^\circ)\) (using a calculator, \(\sin(71^\circ)\approx0.9455\)). Then, \(x=\frac{31}{0.9455}\approx32.8\).

Answer:

32.8