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The problem is asking you to determine two things about the hot-air balloon:

1. The total speed \(v\) of the balloon.
2. The angle \( \theta \) the balloon makes with the horizontal.

### Given Information:
- The balloon is rising vertically at 20 feet per second.
- The wind is blowing horizontally at 5 feet per second.

This forms a right triangle with the vertical velocity (20 ft/sec) as one leg, the horizontal velocity (5 ft/sec) as the other leg, and the hypotenuse being the total speed \(v\) of the balloon.

### Step 1: Calculate the Speed \(v\)
The speed \(v\) is the hypotenuse of the right triangle, so we can use the Pythagorean theorem:
\[
v = \sqrt{(20)^2 + (5)^2}
= \sqrt{400 + 25}
= \sqrt{425}
\approx 20.6 \, \text{ft/sec}
\]
Rounded to the nearest whole number: \(v \approx 21 \, \text{ft/sec}\).

### Step 2: Calculate the Angle \( \theta \)
We can use trigonometry to find the angle. The angle \( \theta \) is the angle between the balloon's path and the horizontal. Using the tangent function:
\[
\tan(\theta) = \frac{\text{opposite (vertical speed)}}{\text{adjacent (horizontal speed)}} = \frac{20}{5} = 4
\]
Now, to find \( \theta \), take the arctangent (inverse tangent):
\[
\theta = \tan^{-1}(4) \approx 75.96^\circ
\]
Rounded to the nearest whole number: \( \theta \approx 76^\circ \).

### Final Answer:
- The balloon’s speed is approximately **21 ft/sec**.
- The angle \( \theta \) it makes with the horizontal is approximately **76 degrees**.

Would you like further details or have any questions?

Here are five related questions you can explore:
1. How would the angle change if the wind speed increased?
2. What would happen to the speed if the vertical ascent rate decreased?
3. Can you calculate the horizontal distance traveled by the balloon after 10 seconds?
4. How do you solve for speed and angle if the horizontal wind speed is unknown?
5. How does the Pythagorean theorem apply to real-world velocity problems?

**Tip:** Always use the Pythagorean theorem for right triangles when combining perpendicular velocities!

A hot-air balloon is rising vertically 20 ft/sec while the wind is blowing horizontally at 5 ft/sec. Find the speed v of the balloon and the angle θ it makes with the horizontal.

This problem demonstrates how to find the speed and angle of a hot-air balloon rising vertically while the wind blows horizontally. It applies the Pythagorean theorem and tangent function to solve for the balloon's speed and angle with the horizontal.

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