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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.

Sagot :

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:

[tex]\sin\theta=\frac{OS}{H}.[/tex]

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

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

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

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