Question

an electrician leans an extension ladder against the outside wall of a house so that it reaches an electric box 24 feet up. the ladder makes an angle of 80° 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 on the wall (opposite side to the given angle) is 24 feet and the length of the ladder is the hypotenuse $x$. The angle between the ladder and the ground is $80^{\circ}$. We use the sine function since $\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}$. So, $\sin(80^{\circ})=\frac{24}{x}$.

Step2: Solve for $x$

Cross - multiply to get $x\sin(80^{\circ}) = 24$. Then $x=\frac{24}{\sin(80^{\circ})}$.
We know that $\sin(80^{\circ})\approx0.9848$. So, $x=\frac{24}{0.9848}\approx24.4$.

Answer:

24.4 feet