from a balloon 1172 feet high, the angle of depression to the ranger headquarters is 74.6 degrees. how far is the headquarters from a point on the ground directly below the balloon to the nearest foot

To solve this problem, we can use trigonometry. The key here is the angle of depression and the right triangle formed by the height of the balloon, the distance between the headquarters and the point on the ground directly below the balloon, and the line of sight from the balloon to the headquarters.

Given:
- Height of the balloon = 1172 feet
- Angle of depression = 74.6°

We are looking for the horizontal distance between the point directly below the balloon and the ranger headquarters, which is the adjacent side of the right triangle.

Using the tangent function:

\[
\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}
\]

In this case:
- \(\theta = 74.6^\circ\)
- Opposite side = 1172 feet
- Adjacent side = distance we need to find

Rearranging the formula to solve for the adjacent side (distance):

\[
\text{adjacent} = \frac{\text{opposite}}{\tan(\theta)} = \frac{1172}{\tan(74.6^\circ)}
\]

Now, calculate:

\[
\text{adjacent} = \frac{1172}{\tan(74.6^\circ)} \approx \frac{1172}{3.594} \approx 326 feet
\]

So, the distance between the point directly below the balloon and the ranger headquarters is approximately **326 feet**.

Would you like any more details or clarifications?

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Here are 5 related questions to explore further:
1. What is the angle of elevation from the ranger headquarters to the balloon?
2. How would the distance change if the angle of depression were increased or decreased?
3. What trigonometric function would we use if we were given the horizontal distance and needed to find the height?
4. How would the problem change if the balloon's height was in meters instead of feet?
5. What would the result be if the angle of depression were exactly 90°?

**Tip:** When solving word problems involving angles of depression or elevation, always visualize or sketch the triangle to help determine the correct trigonometric ratio to use.

From a balloon 1172 feet high, the angle of depression to the ranger headquarters is 74.6 degrees. How far is the headquarters from a point on the ground directly below the balloon to the nearest foot?

Solve a real-world trigonometry problem involving a balloon 1172 feet high and an angle of depression of 74.6 degrees to determine the horizontal distance to the ranger headquarters. Learn to apply the tangent function in right triangles, suitable for students in Grades 9-11.

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