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 63° with the ground. find the length of the ladder. round your answer to the nearest hundredth of a foot if necessary.

Explanation:

Step1: Set up the trigonometric relationship

We know that in a right - triangle (formed by the wall, the ground, and the ladder), the sine function is defined as \(\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}\). Here, the height of the electric box (opposite side) \(y = 31\) feet, the angle \(\theta=63^{\circ}\), and the length of the ladder is the hypotenuse \(x\). So, \(\sin(63^{\circ})=\frac{31}{x}\).

Step2: Solve for \(x\)

We can rewrite the equation from Step 1 as \(x=\frac{31}{\sin(63^{\circ})}\).
Since \(\sin(63^{\circ})\approx0.891\), then \(x=\frac{31}{0.891}\).

$$x=\frac{31}{0.891}\approx34.79$$

Answer:

The length of the ladder is approximately \(34.79\) feet.