Sure, let's solve this step by step using trigonometry.
### Step 1: Understand the problem
We have a right triangle formed by the ladder, the wall, and the ground. The ladder forms the hypotenuse, the distance from the wall to the base of the ladder is one leg, and the height where the ladder touches the wall is the other leg.
Given:
- Hypotenuse (ladder length) = 18 feet
- Adjacent side (distance from the wall to the base of the ladder) = 5.5 feet
We need to find the angle θ that the ladder makes with the ground.
### Step 2: Use cosine formula
In trigonometry, the cosine of an angle in a right triangle is defined as the ratio of the length of the adjacent side to the hypotenuse. This can be expressed as:
[tex]\[ \cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}} \][/tex]
### Step 3: Plug in the values
Here, the adjacent side is 5.5 feet and the hypotenuse is 18 feet.
[tex]\[ \cos(\theta) = \frac{5.5}{18} \][/tex]
### Step 4: Calculate the cosine value
[tex]\[ \cos(\theta) = \frac{5.5}{18} \approx 0.3056 \][/tex]
### Step 5: Find the angle θ
To find the angle θ, we need to take the inverse cosine (also known as arccosine) of 0.3056.
[tex]\[ \theta = \cos^{-1}(0.3056) \][/tex]
Using a calculator, we find:
[tex]\[ \theta \approx 72.24^\circ \][/tex]
### Conclusion
The ladder should make an angle of approximately [tex]\( 72.24^\circ \)[/tex] with the ground.
