Question

Firemen are climbing a 65-foot ladder to the top of a building. If the foot of the ladder is 56 feet away from the building, calculate the angle that the ladder makes with the ground.

Answer 1

To find the angle that the ladder makes with the ground, we can use the trigonometric relationship involving the sine, cosine, or tangent functions. In this case, we can use the cosine function, which relates the adjacent side and hypotenuse of a right triangle.

1. **Identify the sides of the triangle**:
- The length of the ladder (hypotenuse) is \( 65 \) feet.
- The distance from the foot of the ladder to the building (adjacent side) is \( 56 \) feet.

2. **Use the cosine function**:
\[
\cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}}
\]
Substituting the known values:
\[
\cos(\theta) = \frac{56}{65}
\]

3. **Calculate \(\theta\)**:
To find \(\theta\), we will take the arccosine (inverse cosine) of the value calculated:
\[
\theta = \cos^{-1}\left(\frac{56}{65}\right)
\]

4. **Perform the calculation**:
First, calculate \( \frac{56}{65} \):
\[
\frac{56}{65} \approx 0.8615
\]

Now, find the arccosine:
\[
\theta \approx \cos^{-1}(0.8615)
\]
Using a calculator:
\[
\theta \approx 30.34^\circ
\]

Thus, the angle that the ladder makes with the ground is approximately \( \theta \approx 30.34^\circ \).