Part of a highway is to be cantilevered out from a mountainside, as shown. The width of the highway is 22 m, and the angle of the mountain slope at the road measures 51°. An 18 m beam needs to be installed to support the highway. Calculate possible distances, downhill from the highway, where the support post could be fastened.
What distance would you recommend? Explain.
To calculate the possible distances, we first need to use trigonometry to find the height of the mountain slope where the beam will be attached.
Here's the calculation:
height = 22m * sin(51°) = 22m * 0.788 = 17.336 m
Now, we can find the possible distances from the highway where the support post could be fastened using the Pythagorean theorem. Let's denote the distance as x:
x^2 + 17.336^2 = 18^2
x^2 + 300.3257 = 324
x^2 = 23.6743
x ≈ 4.87 m
Therefore, the support post could be fastened at approximately 4.87 meters downhill from the highway.
I would recommend this distance, as it provides sufficient support and keeps the beam at a reasonable distance from the highway, ensuring the safety and stability of the structure.
Part of a highway is to be cantilevered out from a mountainside, as shown. The width of the highway is 22 m, and the angle of the mountain slope at the road measures 51°. An 18 m beam needs to be installed to support the highway. Calculate possible distances, downhill from the highway, where the support post could be fastened.
What distance would you recommend? Explain.
To calculate the possible distances where the support post could be fastened downhill from the highway, we start by determining the horizontal distance from the edge of the highway to the point where the beam will be attached to the mountain.
Since the width of the highway is 22m and the angle of the mountain slope is 51°, we can use trigonometry to find the horizontal distance:
horizontal distance = 22m * cos(51°) ≈ 13.86m
Next, we can use the Pythagorean theorem to find the possible distances from the highway where the support post could be fastened. Let's denote the distance as x:
x^2 + 18^2 = 13.86^2
x^2 + 324 = 192.0996
x^2 = 192.0996 - 324
x^2 ≈ -131.9004 (This negative value indicates that it is not possible to find a feasible location for the support post within the given constraints.)
Since the resulting equation gives a negative value, it implies that there is no feasible location to fasten the support post downhill from the highway with the given dimensions and constraints.
Therefore, based on the calculation, there is no recommended distance as there is no valid solution for where the support post could be fastened in this scenario. Further assessment or adjustments to the dimensions may be necessary to find a suitable location for the support post.