an electrician leans an extension ladder agai...
an electrician leans an extension ladder against the outside wall of a house so that it reaches an electric box 28 feet up. the ladder makes an angle of 71° with the ground. find the length of the ladder. round your answer to the nearest hundredth of a foot if necessary.
Answer
# Answer:
\(29.70\) feet
# Explanation:
## Step1: Identify the trigonometric relationship
We have a right - triangle where the height (opposite side to the given angle) \(y = 28\) feet and the angle \(\theta=71^{\circ}\), and we want to find the length of the ladder (hypotenuse) \(x\). Using the sine function \(\sin\theta=\frac{y}{x}\).
## Step2: Solve for \(x\)
Since \(\sin\theta=\frac{y}{x}\), we can rewrite it as \(x = \frac{y}{\sin\theta}\). Substituting \(y = 28\) and \(\theta = 71^{\circ}\), and knowing that \(\sin(71^{\circ})\approx0.9455\), we get \(x=\frac{28}{\sin(71^{\circ})}=\frac{28}{0.9455}\approx29.70\) feet.