To find the distance from the balloon to its launch place, we will use trigonometry, specifically the concept of the angle of depression. Here we have:
- Altitude: 180 feet (This is the height of the balloon above the ground)
- Angle of Depression: 16°
The angle of depression is the angle formed by a horizontal line and the line of sight down to an object. In this scenario, the line of sight runs to the launch place of the balloon.
### Step-by-Step Solution:
1. Understand the Relationship:
The angle of depression from the balloon to its launch place corresponds to the angle of elevation from the launch place to the balloon. This is because the horizontal line through the balloon and the ground form alternate interior angles with the line of sight, which are congruent.
2. Trigonometric Ratio:
The relationship we are interested in involves the tangent function. The tangent of an angle in a right triangle is the ratio of the length of the opposite side to the length of the adjacent side.
[tex]\[
\tan(\text{angle}) = \frac{\text{opposite}}{\text{adjacent}}
\][/tex]
3. Set Up the Equation:
Here:
- The angle is [tex]\(16^\circ\)[/tex]
- The opposite side is the altitude of the balloon (180 feet)
- The adjacent side is the horizontal distance [tex]\(d\)[/tex] from the balloon to its launch place, which we need to find.
So,
[tex]\[
\tan(16^\circ) = \frac{180 \text{ feet}}{d}
\][/tex]
4. Solve for [tex]\(d\)[/tex]:
Rearrange the equation to solve for [tex]\(d\)[/tex]:
[tex]\[
d = \frac{180 \text{ feet}}{\tan(16^\circ)}
\][/tex]
5. Calculate the Tangent and Distance:
Use a calculator to find [tex]\(\tan(16^\circ)\)[/tex]:
[tex]\[
\tan(16^\circ) \approx 0.2867
\][/tex]
Now calculate [tex]\(d\)[/tex]:
[tex]\[
d = \frac{180 \text{ feet}}{0.2867} \approx 628 \text{ feet}
\][/tex]
### Final Answer:
The horizontal distance from the balloon to its launch place is approximately 628 feet.
