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.

Answer 1

Step-by-step explanation

From the statement, we know that:

• the ladder reaches an electric box 31 ft up,

,

• the ladder makes an angle of 63° with the ground.

Using these data, we make the following diagram:

From the diagram, we see that the ladder and the wall form a right triangle with:

• H = hypotenuse = length of the ladder,

,

• OS = opposite side to angle θ = 31 ft,

,

• θ = angle between the ladder and the ground = 63°,

From trigonometry, we have the following trigonometric relation between H, OS and θ:

`\sin\theta=(OS)/(H).`

Replacing the data from above and solving for H, we get:

`\begin{gathered} \sin(63\degree)=(31ft)/(H), \\ H\cdot\sin(63\degree)=31ft, \\ H=(31ft)/(\sin(63\degree))\cong34.79ft. \end{gathered}`Answer

The length of the ladder is 34.79 ft to the nearest hundred.