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Chapter 4: Problem 22

Height A ladder that is 20 feet long leans against the side of a house. The angle of elevation of the ladder is \(80^{\circ} .\) Find the height from the top of the ladder to the ground.

Short Answer

Expert verified
The height from the top of the ladder to the ground is \(20 \times \sin(80^{\circ})\) feet.

Step by step solution

01

Identify the given information

The length of the ladder is given as 20 feet and the angle of elevation is \(80^{\circ}\). What is needed is the height from the top of the ladder to the ground.
02

Realize that we have a right triangle

Since the ladder is leaning against a wall, the wall, the ladder and the ground form a right triangle.
03

Identify the sides of the triangle

The angle of elevation is formed by the ladder and the ground, so the ladder is the hypotenuse. The height from the top of the ladder to the ground is the opposite side.
04

Use the sine function to find the height

Sin of an angle in a right triangle is defined as the length of the opposite side divided by the length of the hypotenuse. In this scenario, the opposite side is the height we want to find. So we rearrange our equation \(\sin (80^{\circ}) = \frac{height}{20ft}\) to solve for the height, giving \(height = \sin (80^{\circ}) \times 20ft\).
05

Calculate the height

Substitute \(80^{\circ}\) into the sine function and multiply the result by 20 to find the height.

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