A ladder 26.8 feet long leans against a building. If the foot of the ladder is 6.53 feet from the base of the building, find the angle the top of the ladder makes with the building.

To find the angle the ladder makes with the building, we can use trigonometric relationships. Since the ladder leans against the building, forming a right triangle with the ground, we can apply the cosine function:

Let:
- \( L = 26.8 \) feet (the length of the ladder, which is the hypotenuse),
- \( A = 6.53 \) feet (the distance from the base of the building, which is the adjacent side to the angle we want to find).

We want to find the angle \( \theta \) that the ladder makes with the building. Using the cosine function:

\[
\cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{6.53}{26.8}
\]

Calculating the cosine:

\[
\cos(\theta) = \frac{6.53}{26.8} \approx 0.2437
\]

Now, find \( \theta \) by taking the inverse cosine:

\[
\theta = \cos^{-1}(0.2437)
\]

\[
\theta \approx 75.9^\circ
\]

So, the angle the ladder makes with the building is approximately \( 75.9^\circ \).

Would you like further details on this solution or have any other questions?

### Related Questions:
1. How would you find the angle if the ladder's length was changed?
2. What if the distance from the building was different? How would that affect the angle?
3. Can you solve for the height of the ladder on the building using trigonometry?
4. How do you use the sine function to find the same angle in this scenario?
5. What real-life situations might require calculating such angles?

### Tip:
When working with right triangles, identifying the adjacent, opposite, and hypotenuse sides in relation to the angle of interest helps in selecting the correct trigonometric function to use.

Learn how to calculate the angle a ladder makes with a building using trigonometry. This problem involves applying the cosine function with a 26.8-foot ladder and a 6.53-foot distance from the base. Suitable for high school students studying trigonometry.

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