To find the height of the flagpole, we can use trigonometry, particularly the tangent function. Here's a step-by-step solution:
1. Identify given values:
- Eye level height above the ground: 5.5 feet
- Distance from the base of the flagpole: 36 feet
- Angle of elevation: 25 degrees
2. Convert the angle of elevation to radians:
To use trigonometric functions, we need to convert degrees to radians. The angle of 25 degrees is approximately 0.4363323129985824 radians.
3. Use 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]
Where:
- The angle is 0.4363323129985824 radians.
- The adjacent side is the distance from the student to the base of the flagpole, which is 36 feet.
- The opposite side is the height of the flagpole above the student's eye level.
4. Calculate the height opposite from the student's eyes:
Using the tangent function:
[tex]\[
\text{{height opposite from eyes}} = \tan(0.4363323129985824) \times 36
\][/tex]
Performing the calculation, we get:
[tex]\[
\text{{height opposite from eyes}} \approx 16.78707569357995 \text{{ feet}}
\][/tex]
5. Add the student's eye level height to find the total height of the flagpole:
The total height of the flagpole includes the student's eye level height and the height calculated from the tangent function:
[tex]\[
\text{{total flagpole height}} = 5.5 + 16.78707569357995
\][/tex]
6. Calculate the total height:
[tex]\[
\text{{total flagpole height}} \approx 22.28707569357995 \text{{ feet}}
\][/tex]
Therefore, the height of the flagpole is approximately 22.29 feet.
