To find the height of the ladder, we can use trigonometry. Specifically, we can use the tangent function, which relates the angle of elevation to the opposite side (height of the ladder) and the adjacent side (distance from the wall).
Here's the step-by-step solution:
1. Identify the given information:
- The angle of elevation ([tex]\( \theta \)[/tex]) of the ladder is 19 degrees.
- The distance from the wall (adjacent side, [tex]\( d \)[/tex]) is 12 feet.
2. Use the tangent function, which is defined as:
[tex]\[
\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}
\][/tex]
In this case, the opposite side is the height of the ladder ([tex]\( h \)[/tex]) and the adjacent side is the distance from the wall ([tex]\( d \)[/tex]).
3. Rearrange the formula to solve for the height ([tex]\( h \)[/tex]):
[tex]\[
h = d \times \tan(\theta)
\][/tex]
4. Convert the angle from degrees to radians since trigonometric functions typically use radians in calculations.
5. Calculate the tangent of 19 degrees and then multiply by 12 feet.
6. The result should be rounded to the nearest tenth. Given the result is 4.1 feet after performing these steps.
Therefore, the height of the ladder is:
[tex]\[
\boxed{4.1 \text{ ft}}
\][/tex]
