Certainly! Let's solve this problem step-by-step.
1. Understanding the Problem:
- We have a hot air balloon that is 21 feet off the ground.
- There is a rope tied to the ground at a [tex]$30^\circ$[/tex] angle with the ground.
- We need to calculate the horizontal distance from the point directly underneath the balloon to the point where the second rope (tied at the [tex]$30^\circ$[/tex] angle) meets the ground.
2. Trigonometric Relationship:
- Given the height and the angle, we can use trigonometry to find the horizontal distance.
- We will use the tangent function, which relates the opposite side (height) to the adjacent side (horizontal distance) in a right triangle:
[tex]\[
\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}
\][/tex]
- Here, the opposite side is the height of the balloon (21 feet), and the adjacent side is the distance we want to find.
3. Setting Up the Equation:
- Let [tex]\( \theta \)[/tex] be the angle of the rope with the ground, which is [tex]$30^\circ$[/tex].
- Let [tex]\( h \)[/tex] be the height of the balloon, which is 21 feet.
- Let [tex]\( d \)[/tex] be the horizontal distance we are solving for.
- Using the tangent function:
[tex]\[
\tan(30^\circ) = \frac{21}{d}
\][/tex]
4. Solving for the Distance:
- Rearrange the equation to solve for [tex]\( d \)[/tex]:
[tex]\[
d = \frac{21}{\tan(30^\circ)}
\][/tex]
- The value of [tex]\( \tan(30^\circ) \)[/tex] in degrees is approximately 0.5774.
- Plugging this value in, we get:
[tex]\[
d = \frac{21}{0.5774} \approx 36.3731
\][/tex]
5. Rounding to the Nearest Hundredth:
- The calculated distance is approximately 36.3731 feet.
- Rounding this to the nearest hundredth, we get 36.37 feet.
6. Conclusion:
- The horizontal distance between the point directly underneath the balloon and the point where the second rope meets the ground, rounded to the nearest hundredth of a foot, is [tex]\( \boxed{36.37} \)[/tex] feet.
