A graded ramp is to be built to a barn loft. The ramp is to be inclined at an angle of 17°. The floor of the loft is 5 m above ground level. Find the length of the ramp, in 2 decimal places.

To find the length of the ramp, we can use trigonometry. The ramp, the height from the ground to the loft, and the ground form a right triangle.

Given:
- The angle of inclination \(\theta = 17^\circ\)
- The height \(h = 5\) m (opposite side to the angle)

We need to find the length of the hypotenuse \(L\), which represents the length of the ramp.

We can use the sine function, which is defined as:
\[
\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{h}{L}
\]
Rearranging to solve for \(L\):
\[
L = \frac{h}{\sin(\theta)}
\]
Substituting the values:
\[
L = \frac{5}{\sin(17^\circ)}
\]

Now, calculate the value of \(L\):

\[
L \approx \frac{5}{\sin(17^\circ)} \approx \frac{5}{0.2924} \approx 17.10 \text{ m}
\]

So, the length of the ramp is approximately \(17.10\) meters.

Would you like further details or have any questions?

### Related Questions:
1. How would the length change if the angle was increased to 25°?
2. What is the horizontal distance covered by the ramp?
3. What would be the length of the ramp if the loft height was 7 meters instead?
4. How does the sine function relate to other trigonometric functions in this context?
5. What is the angle of inclination if the ramp length is 20 meters?
6. What is the cosine of the angle of inclination, and how does it help in finding the ramp's horizontal distance?
7. Can we solve this problem using the tangent function? If so, how?
8. What would be the effect of decreasing the angle to 10° on the ramp length?

**Tip:** In trigonometry, the sine function is especially useful for finding unknown side lengths when dealing with right triangles and angles of elevation or depression.

Learn how to calculate the length of a ramp inclined at 17 degrees to reach a loft 5 meters above ground. This problem involves trigonometry, specifically the sine function, and is suitable for students in Grades 9-12.

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