The road is centered on the axis of symmetry and the transport truck must stay on its own side of the road. Can the transport truck safely pass through the bridge?  Justify your decision using a clearly constructed mathematical model.

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The general equation of a parabola where the axis of symmetry is the y-axis has the form y-k=ax², where k is the y-intercept, and the vertex is (0,k). The highest point of the arch is 10m above the ground and this point is the vertex, so k=10. We have y=ax²+10. The ground width of the arch is 16m, which means that the x-intercepts are -8 and 8. Therefore x²=64 and y=0, so:

0=64a+10. Therefore, a=-10/64=-5/32, making y=10-5x²/32.

The truck must stay on its correct side of the road so, since we know its height is 3.5m, the distance between this height and the arch wall must exceed the width of the truck (5m) at this height, otherwise the truck cannot keep to its own side of the road and avoid hitting the arch wall.

The height corresponds to the y value, so we can calculate the corresponding x value:

3.5=10-5x²/32, x²=32×(10-3.5)/5=32×1.3=41.6m².

x=6.45m approx.

The truck is 5m wide so it has clearance of about 1.45m and will be able to pass under the arch.

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