Let's break down the problem:

Problem:

Tallulah leans a 22-foot ladder against a wall. The ladder reaches 19.9 feet up the wall. We need to find the angle that the ladder forms with the ground, rounded to the nearest tenth of a degree.

Solution:

This problem involves a right triangle, where:

• The hypotenuse ($c$) is the length of the ladder, 22 feet.
• The opposite side ($a$) is the height the ladder reaches on the wall, 19.9 feet.
• The angle $\theta$ is the one formed between the ladder and the ground, and we need to find this angle.

Using trigonometry, specifically the sine function, we have: $\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{19.9}{22}$

Now, calculate $\sin(\theta)$: $\sin(\theta) = \frac{19.9}{22} \approx 0.9045$

To find $\theta$, take the inverse sine ($\sin^{-1}$) of 0.9045: $\theta = \sin^{-1}(0.9045)$

Using a calculator: $\theta \approx 64.7^\circ$

Answer:

The angle that the ladder forms with the ground is approximately 64.7 degrees.

Would you like more details or have any other questions? Here are some related questions for further practice:

1. If the ladder were 25 feet long and still reached 19.9 feet up the wall, what angle would it form with the ground?
2. What would be the horizontal distance from the base of the wall to the bottom of the ladder for this problem?
3. How would the angle change if the ladder reached a higher point on the wall?
4. Can you find the angle using cosine instead of sine in this case? Why or why not?
5. What is the general formula for finding the angle in such ladder problems?

Tip: When solving trigonometric problems, always make sure your calculator is set to the correct mode (degrees or radians) based on what the problem asks for.