To find the height of the platform given a ramp length and angle of elevation, we can use trigonometry, specifically the sine function. Here is a step-by-step solution:
1. Identify the known values:
- Length of the ramp (hypotenuse) = 23.9 feet
- Angle of elevation = 9.1°
2. Understand the trigonometric relationship:
- The sine of an angle in a right triangle is defined as the ratio of the length of the opposite side (which is the height of the platform in this case) to the length of the hypotenuse.
- Formula: [tex]\( \sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} \)[/tex]
3. Set up the equation:
- Here, [tex]\(\theta = 9.1°\)[/tex]
- Hypotenuse = 23.9 feet
- Let [tex]\( h \)[/tex] be the height of the platform.
- Therefore, [tex]\( \sin(9.1°) = \frac{h}{23.9} \)[/tex]
4. Solve for the height [tex]\( h \)[/tex]:
- [tex]\( h = 23.9 \times \sin(9.1°) \)[/tex]
5. Calculate [tex]\( \sin(9.1°) \)[/tex]:
- Using a calculator or sine tables, find the sine of 9.1°.
6. Compute the height:
- Substitute the sine value into the equation and multiply:
- [tex]\( h \approx 23.9 \times 0.158 \)[/tex]
7. Perform the multiplication:
- The computed height is approximately 3.7799778073821564 feet.
8. Round to the nearest tenth:
- When rounded to the nearest tenth, the height is approximately 3.8 feet.
So, the height of the platform is 3.8 feet when rounded to the nearest tenth of a foot.
