To find the distance [tex]\(X\)[/tex] from Roman's feet to the base of the flag pole, we can use trigonometric relationships in a right triangle. The problem provides the following information:
- The height of the flag pole, which is 17 meters.
- The angle of elevation from Roman's position to the top of the flag pole, which is 39 degrees.
We can use the tangent function from trigonometry, which is defined as:
[tex]\[ \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} \][/tex]
In our scenario:
- [tex]\(\theta = 39^\circ\)[/tex] is the angle of elevation.
- The "opposite" side of the triangle (which is the height of the flag pole) is 17 meters.
- The "adjacent" side of the triangle (which is the distance from Roman to the base of the flag pole) is what we are solving for, represented by [tex]\(X\)[/tex].
Using the tangent function:
[tex]\[ \tan(39^\circ) = \frac{17}{X} \][/tex]
To solve for [tex]\(X\)[/tex], we rearrange the equation:
[tex]\[ X = \frac{17}{\tan(39^\circ)} \][/tex]
Evaluating this expression, we find that:
[tex]\[ X \approx 20.993 \][/tex]
Rounding the computed distance to the nearest tenth, we get:
[tex]\[ X \approx 21.0 \][/tex]
Thus, the distance from Roman's feet to the base of the flag pole is approximately 21.0 meters.
