To solve this problem, we can use trigonometry. Specifically, we are dealing with a right triangle, where:

• The length of the ladder forms the hypotenuse of the triangle (22 feet).
• The angle between the ladder and the ground is 74 degrees.
• We need to find the height up the wall, which corresponds to the length of the vertical leg of the triangle.

We will use the sine function because it relates the angle to the ratio of the opposite side (height) and the hypotenuse.

The formula is:

$\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}$

Where:

• $\theta = 74^\circ$
• The hypotenuse = 22 feet
• The opposite (height) = ? (this is what we're trying to find)

Rearranging the formula to solve for the height:

$\text{height} = \sin(74^\circ) \times 22$

Using a calculator to find $\sin(74^\circ)$:

$\sin(74^\circ) \approx 0.961261$

Now, calculate the height:

$\text{height} = 0.961261 \times 22 \approx 21.14 \text{ feet}$

So, the ladder reaches approximately 21.14 feet up the wall.

Would you like further details on the steps, or do you have any other questions?

Here are some related questions you could explore:

1. How can we use the cosine function to find the horizontal distance from the base of the ladder to the wall?
2. How does changing the angle affect the height reached by the ladder?
3. What happens if the ladder is 30 feet long instead of 22 feet?
4. How would the problem change if the ladder were placed at a 45-degree angle?
5. What are some real-life applications of right triangle trigonometry in construction?

Tip: When dealing with right triangles and angles, remember that sine, cosine, and tangent functions are key to solving for unknown sides and angles.