To solve this problem, we start by considering a right triangle, where the base length is the distance from the point of observation to the base of the statue, the height of the statue is the opposite side we need to find, and the angle of elevation to the top of the statue is given.
Step-by-Step Solution:
1. Identify the given values:
- Distance from the base of the statue: [tex]\(165\)[/tex] feet (adjacent side in the right triangle)
- Angle of elevation: [tex]\(10\)[/tex] degrees
2. Use trigonometric functions:
- The angle of elevation (10 degrees) is given, and we're dealing with opposite and adjacent sides of a right triangle.
- The appropriate trigonometric function to use is the tangent function, which is defined as:
[tex]\[
\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}
\][/tex]
- Here, [tex]\(\theta = 10\)[/tex] degrees, the opposite side is the height of the statue, and the adjacent side is [tex]\(165\)[/tex] feet.
3. Set up the equation:
- Plug the known values into the tangent function equation:
[tex]\[
\tan(10^\circ) = \frac{\text{height}}{165}
\][/tex]
- Rearrange the equation to solve for the height:
[tex]\[
\text{height} = 165 \times \tan(10^\circ)
\][/tex]
4. Calculate the height of the statue:
- Evaluate the tangent of 10 degrees. This step would usually be done using a calculator to obtain an accurate value.
- Multiply that value by [tex]\(165\)[/tex] feet.
Conclusion:
The height of the statue, based on the given calculations and the trigonometric relationship, is approximately [tex]\(29.09\)[/tex] feet. This is the height of the statue from its base to the top when observed at an angle of elevation of 10 degrees from 165 feet away.
