a 20 foot ladder leans against a wall so that the ladders angle of elevation is 46 degrees. find x, the distance from the base of the ladder to the building

Answer 1

To find the distance \( x \) from the base of the ladder to the wall (building), we can use trigonometric functions. Specifically, we will use the cosine function, which relates the adjacent side to the hypotenuse in a right triangle.

Here are the details:

• The length of the ladder (hypotenuse) is \( 20 \) feet.
• The angle of elevation (the angle between the ground and the ladder) is \( 46^\circ \).

The formula using cosine is:

\[ \cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}} \]

In our case:

\[ \cos(46^\circ) = \frac{x}{20} \]

To find \( x \), we can rearrange the equation:

\[ x = 20 \cdot \cos(46^\circ) \]

Now we can calculate \( x \):

1. Calculate \( \cos(46^\circ) \):

   • Using a calculator, we find \( \cos(46^\circ) \approx 0.6947 \).

2. Multiply by the length of the ladder: \[ x \approx 20 \cdot 0.6947 \approx 13.894 \]

So the distance \( x \) from the base of the ladder to the building is approximately \( 13.89 \) feet.