A 16 ft ladder leans against the side of a house. The top of the ladder is 15 ft off the ground. Find x, the angle of elevation of the ladder. Round your answer to the nearest tenth of a degree.

To find the angle of elevation \( x \) of the ladder, you can use trigonometry. Specifically, the sine function relates the angle of elevation to the length of the ladder (the hypotenuse) and the height of the top of the ladder above the ground (the opposite side of the angle).

The formula for sine is:

\[
\sin(x) = \frac{\text{opposite}}{\text{hypotenuse}}
\]

Here:
- The "opposite" side is 15 feet (the height of the ladder from the ground).
- The "hypotenuse" is 16 feet (the length of the ladder).

Substituting the given values:

\[
\sin(x) = \frac{15}{16}
\]

To find \( x \), take the inverse sine (also called arcsine) of both sides:

\[
x = \sin^{-1}\left(\frac{15}{16}\right)
\]

Now, let's calculate the angle \( x \).

\[
x \approx \sin^{-1}\left(0.9375\right) \approx 69.1^\circ
\]

So, the angle of elevation \( x \) is approximately **69.1°**.

Would you like further details or have any questions?

---

Here are 5 related questions you might consider:

1. How would you find the angle of elevation if the ladder's length were unknown?
2. What would be the angle of elevation if the ladder were only 10 feet tall?
3. How does changing the height of the ladder affect the angle of elevation?
4. Can the angle of elevation ever be 90°? Why or why not?
5. How would the problem change if the ladder were leaning against a different object, like a tree?

**Tip:** When solving right triangle problems, always double-check which trigonometric function (sine, cosine, or tangent) is most appropriate based on the sides you're given.

Learn how to calculate the angle of elevation of a 16 ft ladder leaning against a house with a step-by-step solution. This problem involves using trigonometry and the sine function to find the angle of elevation. Suitable for high school students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation