To determine the length of the ramp, we'll employ trigonometric principles, specifically using the sine function. Here’s a detailed step-by-step solution:
### Step-by-Step Solution:
1. Understand the given values:
- The angle of inclination of the ramp with respect to the horizontal ground: [tex]\( 17^\circ \)[/tex]
- The vertical height from the ground to the loft: [tex]\( 5 \)[/tex] meters
2. Convert the angle to radians:
Trigonometric calculations can be more accurate when done in radians. The conversion formula from degrees to radians is:
[tex]\[
\text{Radians} = \text{Degrees} \times \left(\frac{\pi}{180}\right)
\][/tex]
By converting [tex]\( 17^\circ \)[/tex] to radians:
[tex]\[
17 \times \left(\frac{\pi}{180}\right) \approx 0.2967 \text{ radians}
\][/tex]
3. Set up the trigonometric relationship:
Using the sine function, which relates the angle to the opposite side (height) and the hypotenuse (length of the ramp):
[tex]\[
\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}
\][/tex]
Here, [tex]\(\theta\)[/tex] is the angle of inclination (17° or [tex]\( \approx 0.2967 \text{ radians}\)[/tex]), the opposite side is the height (5 meters), and the hypotenuse is the ramp length ([tex]\(L\)[/tex]).
Thus:
[tex]\[
\sin(0.2967) = \frac{5}{L}
\][/tex]
4. Solve for the hypotenuse (length of the ramp):
Rearrange the formula to isolate [tex]\(L\)[/tex]:
[tex]\[
L = \frac{5}{\sin(0.2967)}
\][/tex]
5. Calculate the sine of the angle:
[tex]\[
\sin(0.2967) \approx 0.2924
\][/tex]
6. Substitute and solve for the ramp length [tex]\(L\)[/tex]:
[tex]\[
L = \frac{5}{0.2924} \approx 17.1015
\][/tex]
7. Round the length of the ramp to 2 decimal places:
[tex]\[
L \approx 17.10
\][/tex]
### Conclusion:
The length of the ramp, when rounded to 2 decimal places, is approximately [tex]\( 17.10 \)[/tex] meters.
