https://file.aiquickdraw.com/mathbot/file/1726539593711d4fnhozt.jpg

Let's break down the problem in the image:

### Given:
- The length of the rope is 680 ft.
- The height of the balloon is \( h \) ft.
- The angle \( \theta \) is formed between the ground and the rope anchoring the balloon.

We are asked to solve the following two parts:

### (a) Express \( \theta \) as a function of \( h \):

Using basic trigonometry, we can observe that the setup forms a right triangle, where:
- The rope length is the hypotenuse of the triangle (680 ft),
- The height \( h \) is the opposite side to angle \( \theta \).

The sine function is defined as:
\[
\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{h}{680}
\]

So, to express \( \theta \) as a function of \( h \):
\[
\theta = \sin^{-1}\left(\frac{h}{680}\right)
\]

### (b) Find the angle \( \theta \) when the balloon is 470 ft high:

We will substitute \( h = 470 \) ft into the equation derived in part (a).

\[
\theta = \sin^{-1}\left(\frac{470}{680}\right)
\]

Let me calculate this value for \( \theta \).The angle \( \theta \) is approximately \( 43.7^\circ \) when the balloon is 470 feet high.

### Summary:
- **(a)** \( \theta = \sin^{-1}\left(\frac{h}{680}\right) \)
- **(b)** When the balloon is 470 ft high, \( \theta \approx 43.7^\circ \).

Do you have any questions or would you like more details on this?

Here are 5 related questions for further exploration:
1. What would the angle \( \theta \) be if the balloon were 600 ft high?
2. How would the formula change if the length of the rope were increased to 700 ft?
3. Can you express the height \( h \) as a function of \( \theta \)?
4. What is the maximum value of \( \theta \), and under what condition would it occur?
5. How would the problem change if the ground were not perfectly flat?

**Tip**: Always remember to double-check if your calculator is set to degrees or radians when performing trigonometric calculations!

A 680-foot rope anchors a hot-air balloon. Express the angle θ as a function of the height h of the balloon and find the angle θ when the height is 470 feet.

Learn how to express the angle θ as a function of the height h in a hot-air balloon problem using trigonometric identities. Calculate the angle when the balloon is 470 feet high. Suitable for students in Grades 9-11.

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