Question
A ladder leans against a wall, forming an angle of 60 degrees with the ground, and its base is 2.5 feet away from the wall. Can you find a similar problem and tell me how to solve it using cosine?
Answers
GPT-4o mini
Certainly! Let's first articulate a similar problem involving a ladder leaning against a wall and then solve it using the cosine function.
### Similar Problem:
Suppose you have a ladder that is 10 feet long. It leans against a wall, forming an angle of \( \theta = 45^\circ \) with the ground. You want to determine how far the base of the ladder is from the wall.
### Solution Using Cosine:
To solve this problem, we can visualize a right triangle formed by the ladder, the wall, and the ground. Here are the steps:
1. **Define the right triangle**:
- The ladder acts as the hypotenuse of the triangle.
- The height (i.e., the distance up the wall) is one leg of the triangle.
- The distance from the base of the ladder to the wall is the other leg of the triangle.
2. **Label the sides**:
- Let \( L = 10 \) feet (the length of the ladder).
- Let \( d \) be the distance from the wall (the base of the ladder).
- Let \( h \) be the height alongside the wall.
3. **Use the cosine function**:
The cosine function relates the angle formed with the ground, the adjacent side (which is the distance from the wall, \( d \)), and the hypotenuse (the ladder length \( L \)):
\[
\cos(\theta) = \frac{\text{Adjacent}}{\text{Hypotenuse}} = \frac{d}{L}
\]
4. **Substitute known values into the equation**:
Given that \( \theta = 45^\circ \) and \( L = 10 \) feet, we can write:
\[
\cos(45^\circ) = \frac{d}{10}
\]
We know that \( \cos(45^\circ) = \frac{\sqrt{2}}{2} \).
5. **Solve for \( d \)**:
\[
\frac{\sqrt{2}}{2} = \frac{d}{10}
\]
Multiply both sides by 10:
\[
d = 10 \cdot \frac{\sqrt{2}}{2} = 5\sqrt{2} \text{ feet}.
\]
6. **Numerical approximation** (if needed):
- \( 5\sqrt{2} \approx 5 \cdot 1.414 = 7.07 \text{ feet} \).
### Conclusion:
For the original problem regarding the ladder forming an angle of \( 60^\circ \) with the ground and being \( 2.5 \) feet away from the wall, you would use similar steps to find the height \( h \) as follows:
Using \( \cos(60^\circ) = \frac{1}{2} \):
\[
\frac{1}{2} = \frac{2.5}{L} \quad \text{(where \( L \) is the length of the ladder)}
\]
Thus:
\[
L = 2.5 \cdot 2 = 5 \text{ feet}.
\]
This method demonstrates how you can apply the cosine function to find various dimensions related to a ladder leaning against a wall.
### Similar Problem:
Suppose you have a ladder that is 10 feet long. It leans against a wall, forming an angle of \( \theta = 45^\circ \) with the ground. You want to determine how far the base of the ladder is from the wall.
### Solution Using Cosine:
To solve this problem, we can visualize a right triangle formed by the ladder, the wall, and the ground. Here are the steps:
1. **Define the right triangle**:
- The ladder acts as the hypotenuse of the triangle.
- The height (i.e., the distance up the wall) is one leg of the triangle.
- The distance from the base of the ladder to the wall is the other leg of the triangle.
2. **Label the sides**:
- Let \( L = 10 \) feet (the length of the ladder).
- Let \( d \) be the distance from the wall (the base of the ladder).
- Let \( h \) be the height alongside the wall.
3. **Use the cosine function**:
The cosine function relates the angle formed with the ground, the adjacent side (which is the distance from the wall, \( d \)), and the hypotenuse (the ladder length \( L \)):
\[
\cos(\theta) = \frac{\text{Adjacent}}{\text{Hypotenuse}} = \frac{d}{L}
\]
4. **Substitute known values into the equation**:
Given that \( \theta = 45^\circ \) and \( L = 10 \) feet, we can write:
\[
\cos(45^\circ) = \frac{d}{10}
\]
We know that \( \cos(45^\circ) = \frac{\sqrt{2}}{2} \).
5. **Solve for \( d \)**:
\[
\frac{\sqrt{2}}{2} = \frac{d}{10}
\]
Multiply both sides by 10:
\[
d = 10 \cdot \frac{\sqrt{2}}{2} = 5\sqrt{2} \text{ feet}.
\]
6. **Numerical approximation** (if needed):
- \( 5\sqrt{2} \approx 5 \cdot 1.414 = 7.07 \text{ feet} \).
### Conclusion:
For the original problem regarding the ladder forming an angle of \( 60^\circ \) with the ground and being \( 2.5 \) feet away from the wall, you would use similar steps to find the height \( h \) as follows:
Using \( \cos(60^\circ) = \frac{1}{2} \):
\[
\frac{1}{2} = \frac{2.5}{L} \quad \text{(where \( L \) is the length of the ladder)}
\]
Thus:
\[
L = 2.5 \cdot 2 = 5 \text{ feet}.
\]
This method demonstrates how you can apply the cosine function to find various dimensions related to a ladder leaning against a wall.