Final answer:
The equation of the parabola that models the road surface is y = -1/640x^2 + 0.4. Using this equation, it is determined that the road surface is 0.1 foot lower than the middle at 8 feet from the center of the road.
Step-by-step explanation:
To find an equation of the parabola that models the road surface, we need to consider that the road's cross-section is a parabola that opens downward (since it's higher in the center). The vertex of the parabola is at the road's highest point, in the center. Since the road is symmetrical, the vertex will be on the y-axis, and we'll place it at the origin for simplicity, which means the vertex is at the point (0, 0.4). The road is 32 feet wide, so it intersects the x-axis at (-16,0) and (16,0).
The general form of a parabola's equation is y = ax^2 + bx + c. Since the parabola is symmetrical about the y-axis, the linear term (represented by 'b' in the equation) is zero. The vertex form of the equation of a parabola is y = a(x - h)^2 + k, where (h,k) is the vertex. Using the vertex (0, 0.4), our equation becomes y = a(x - 0)^2 + 0.4, which simplifies to y = ax^2 + 0.4. We can now use the fact that the parabola passes through the point (16,0) to find 'a'.
Substituting the point into the equation gives us 0 = a(16)^2 + 0.4, which simplifies to -0.4 = 256a. Solving for 'a', we get a = -0.4/256 = -1/640. So, the equation of the parabola is y = -1/640x^2 + 0.4. To find out how far from the center of the road the surface is 0.1 foot lower than the middle, we set y equal to 0.3 (0.4 - 0.1) and solve for x:
0.3 = -1/640x^2 + 0.4
0.1 = -1/640x^2
-64 = x^2
x = ±8
Thus, the road surface is 0.1 foot lower than the middle at 8 feet from the center on either side.