Final answer:
An inequality describing the archway's opening is y < 0.1x^2 + 12. A camper that is 6ft wide and 7ft tall can fit under the arch without crossing the median line, as the minimum height at the edges of the camper's width is 12.9 ft.
Step-by-step explanation:
To address the student's mathematics question involving the quadratic function y = 0.1x2 + 12, we should start by discussing each part of the question:
a) Inequality for the archway's opening:
Since the archway is over a road and symmetrically cut out, we are interested in the y-values that are above the road surface (y ≥0). The inequality that describes the opening of the archway where the vehicle can safely pass would be y < 0.1x2 + 12.
c) Fitting a camper under the archway:
To determine if a camper that is 6ft wide and 7ft tall can fit without crossing the median line, we can find the height of the opening at the points x=±3 (representing the edges of the camper centered at the median line x=0). Plugging these values into our function gives us y = 0.1(3)2 + 12 = 0.9 + 12 = 12.9 ft for height at the edges. This is higher than the 7ft height of the camper, so yes, it can fit under the archway without crossing the median line, provided we are just considering the edges and not the curvature of the camper or arch.