Avery leans a 24-foot ladder against a wall so that it forms an angle of 80degrees∘ with the ground. How high up the wall does the ladder reach? Round your answer to the nearest tenth of a foot if necessary.

To solve this problem, you can use trigonometry, specifically the sine function. The sine of an angle in a right triangle is the ratio of the length of the side opposite the angle to the length of the hypotenuse.

Given:
- The length of the ladder (hypotenuse) \( c = 24 \) feet
- The angle between the ladder and the ground \( \theta = 80^\circ \)

We want to find the height \( h \) up the wall, which is the side opposite the angle.

The sine function is defined as:
\[
\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{h}{c}
\]

So, we can solve for \( h \):
\[
h = c \cdot \sin(\theta)
\]

Substituting the values:
\[
h = 24 \cdot \sin(80^\circ)
\]

Now, calculate \( h \):
\[
h \approx 24 \cdot 0.9848 \approx 23.6 \text{ feet}
\]

So, the ladder reaches approximately **23.6 feet** up the wall.

Would you like more details or have any questions?

Here are some related questions you might find interesting:
1. What is the horizontal distance from the base of the ladder to the wall?
2. How would the height change if the ladder were at a 70-degree angle instead?
3. What if the ladder was 30 feet long? How high up would it reach at the same angle?
4. What is the relationship between the angle and the height reached for a fixed ladder length?
5. How would the height change if the angle were 90 degrees?

**Tip:** When dealing with angles and lengths in right triangles, the sine, cosine, and tangent functions are your best tools for finding missing sides.

Learn how to calculate the height reached by a 24-foot ladder leaning against a wall at an angle of 80 degrees using trigonometry. This problem involves applying the sine function to find the opposite side of a right triangle. 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