To solve this problem, we can use trigonometry. Specifically, we'll use the sine function, which relates the lengths of the sides of a right-angled triangle to its angles.
Here are the steps to solve the problem:
1. **Understanding the scenario:**
We have a ladder leaning against a wall forming a right-angled triangle with the wall and the ground. The length of the ladder is the hypotenuse of the triangle, the height up the wall is the opposite side to the angle made by the ladder with the ground, and the base is the distance from the wall along the ground to the bottom of the ladder.
2. **Identifying the known values:**
- The length of the ladder (hypotenuse) is 10 feet.
- The angle between the ladder and the ground is 65°.
3. **Choosing the right trigonometric function:**
Since we have the hypotenuse and need to find the opposite side (the height up the wall), we will use the sine function.
The sine function is defined as follows:
\[\sin(\text{angle}) = \frac{\text{opposite side}}{\text{hypotenuse}}\]
4. **Setting up the equation:**
Let's denote the height that the ladder reaches up the wall as \( h \). The equation, using the sine function, will be:
\[\sin(65°) = \frac{h}{10 \text{ feet}}\]
5. **Solving for the height (opposite side):**
To find \( h \), we multiply both sides of the equation by the length of the ladder:
\[h = 10 \text{ feet} \times \sin(65°)\]
6. **Calculating the sine of 65°:**
You can either use a scientific calculator or a trigonometry table to find the value of \( \sin(65°) \).
7. **Performing the calculation:**
After you've found the sine of 65°, you multiply it by 10 feet to get the height.
8. **Result:**
Once you perform this multiplication, you will have the distance the ladder reaches up the wall. Remember to keep your calculator in degree mode since the angle provided is in degrees.
Using a calculator to determine \( \sin(65°) \), we get approximately 0.9063. By multiplying this by the ladder's length, we find the height:
\[h = 10 \text{ feet} \times 0.9063 \approx 9.063 \text{ feet}\]
So, the top of the ladder is approximately 9.063 feet up the wall.
