To determine the length of the slide, we need to use trigonometry. Specifically, we will use the sine function which relates the angle of elevation, the height of the slide, and the length of the slide.
Let's break this problem into detailed steps:
1. Identify the given values and the required value:
- Height of the slide (opposite side in trigonometry), [tex]\( h = 12 \)[/tex] feet
- Angle of elevation, [tex]\( \theta = 25^\circ \)[/tex]
- We need to find the length of the slide (the hypotenuse in trigonometry)
2. Recall the definition of sine in a right triangle:
[tex]\( \sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} \)[/tex]
3. Set up the equation for the sine function:
[tex]\[
\sin(25^\circ) = \frac{12}{\text{hypotenuse}}
\][/tex]
4. Solve for the hypotenuse:
[tex]\[
\text{hypotenuse} = \frac{12}{\sin(25^\circ)}
\][/tex]
5. Convert the angle from degrees to radians. In trigonometric calculations involving sine, cosine, and tangent, angles should typically be in radians. The conversion factor is:
[tex]\[
1 \text{ degree} = \frac{\pi}{180} \text{ radians}
\][/tex]
Therefore,
[tex]\[
25^\circ \times \frac{\pi}{180} \approx 0.436 \text{ radians}
\][/tex]
6. Calculate [tex]\( \sin(25^\circ) \)[/tex] after converting the angle to radians:
[tex]\[
\sin(0.436) \approx 0.4226
\][/tex]
7. Substitute [tex]\( \sin(25^\circ) \)[/tex] into the equation:
[tex]\[
\text{hypotenuse} = \frac{12}{0.4226} \approx 28.394
\][/tex]
8. Round the hypotenuse to the nearest tenth:
[tex]\[
\text{hypotenuse} \approx 28.4
\][/tex]
Therefore, the length of the slide, when rounded to the nearest tenth, is 28.4 feet.
