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Chapter 1: Problem 42

A ladder \(9.00 \mathrm{~m}\) long leans against the side of a building. If the ladder is inclined at an angle of \(75.0^{\circ}\) to the horizontal, what is the horizontal distance from the bottom of the ladder to the building?

Short Answer

Expert verified
The horizontal distance from the bottom of the ladder to the building is approximately 2.33 meters.

Step by step solution

01

Understand the Problem

The first step is to visualize the problem. The ladder leaning against the building, the ground, and the side of the building forms a right triangle. The ladder acts as the hypotenuse of the triangle with a length of 9 meters. The angle between the ground (adjacent side) and the ladder (hypotenuse) is 75 degrees.
02

Apply Trigonometric Identity

In trigonometry, we know that the cosine of an angle in a right triangle is equal to the adjacent side divided by the hypotenuse. The adjacent side, in this case, is the ground which is the horizontal distance from the bottom of the ladder to the building that we're trying to find, and the hypotenuse is the ladder itself (9 meters). So, we can represent this relation as: cos(75) = adjacent/9.
03

Solve for the Adjacent Side (Horizontal Distance)

To find the adjacent side (which is the horizontal distance from the bottom of the ladder to the building), we rearrange the formula from step 2 to look like this: adjacent = cos(75) * 9. Solving this using a calculator, we find that the horizontal distance is approximately 2.33 meters.

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