An electrician leans an extension ladder against the outside wall of a house so that it reaches an electric box 27 feet up. The ladder makes an angle of \( 62^{\circ} \) with the ground. Find the length of the ladder. Round your answer to the nearest tenth of a foot if necessary.

To find the length of the ladder, we can use the sine function from trigonometry. The sine of the angle is defined as the ratio of the opposite side (height of the electric box) over the hypotenuse (length of the ladder). Using the formula: \[ \sin(62^{\circ}) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{27}{L} \] Where \(L\) is the length of the ladder. Rearranging gives: \[ L = \frac{27}{\sin(62^{\circ})} \] Calculating \(\sin(62^{\circ}) \approx 0.8870\): \[ L \approx \frac{27}{0.8870} \approx 30.5 \text{ feet} \] Thus, the length of the ladder is approximately **30.5 feet**. --- In the realm of physics and engineering, understanding angles and measurements is crucial for ensuring safety. Proper ladder placement can prevent accidents, which is why angles such as \(62^{\circ}\) are often recommended for optimal stability. When setting up a ladder, remember that the base should be about one-fourth the height of the wall being reached. For a height of 27 feet, you'd want the base about 6.75 feet away from the wall to keep things safe and stable. Happy climbing, and always be cautious!