Question

Rashaad leans a 16-foot ladder against a wall so that it forms an angle of 66degrees
∘
with the ground. What’s the horizontal distance between the base of the ladder and the bottom of the wall? Round your answer to the nearest tenth of a foot if necessary.

Answer 1

To find the horizontal distance between the base of the ladder and the bottom of the wall, we can use trigonometric functions, specifically the cosine function.

The setup of the problem can be visualized as a right triangle where:
- The ladder represents the hypotenuse (16 feet).
- The horizontal distance we are trying to find (the base) is adjacent to the angle (66 degrees).
- The wall represents the opposite side of the triangle.

Using the cosine function, which relates the adjacent side to the hypotenuse, we have:

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

In our case:
\[
\theta = 66^\circ,
\]
\[
\text{hypotenuse} = 16 \text{ feet},
\]
\[
\text{adjacent} = x \text{ (the horizontal distance we want to find)}.
\]

Thus, we can write:
\[
\cos(66^\circ) = \frac{x}{16}.
\]

Now, we can solve for \( x \):

\[
x = 16 \cdot \cos(66^\circ).
\]

Calculating \( \cos(66^\circ) \):

Using a calculator or cosine table:
\[
\cos(66^\circ) \approx 0.4067.
\]

Now plug this value back into the equation:

\[
x = 16 \cdot 0.4067 \approx 6.5072.
\]

Rounding to the nearest tenth of a foot, we get:

\[
x \approx 6.5.
\]

So the horizontal distance between the base of the ladder and the bottom of the wall is approximately **6.5 feet**.