A railroad trestle spans a gorge 80 feet wide and connects two cliffs at heights of 112 and 172 feet above the bottom of the gorge. A train is crossing this gorge from the higher cliff to the lower. When the front of the train has traveled $\frac{3}{4}$ of the trestle's length, how many feet is it above the bottom of the gorge?
Since the railroad trestle is a straight path connecting the two cliffs, we know that after the train has traveled $3/4$ of the trestle's length, the train has traveled both $3/4$ the horizontal distance between the two cliffs and the $3/4$ of the vertical distance between the cliffs' heights. The difference between the heights of the two cliffs is $172-112=60$ feet. $3/4$ of that difference is $45$ feet. Since the train started from the higher cliff and was moving toward the lower cliff, it has descended $45$ feet from its original altitude of $172$ feet. Therefore, the train is now $172-45=\boxed{127}$ feet above the bottom of the gorge.