Answer:
[tex]\boxed{\text{3.74 m}}[/tex]
Step-by-step explanation:
This is equivalent to finding the equation of a parabola from three points.
Let's call the ground-level points (-3.5, 0) and (3.5, 0).
The axis of symmetry will be the y-axis, so the vertex is at (0, 7).
Part 1. Find the equation of the parabola
The vertex form of the equation for a parabola is
y = a(x - h)² + k
where (h, k) is the vertex of the parabola.
If the vertex is at (0, 7),
h = 0 and k = 7
The equation is
y = ax² + 7
Insert the point (0, -3.5)
0 = a(0 + 3.5)² + 7
0 = a(3.5)² + 7
0 = 12.25a + 7
-7 = 12.25a
a = -0.5714
The equation in vertex form is
y = -0.5714x² + 7
Part 2. Calculate the width when y = 5
5 = -0.5714x² + 7
-2 = -0.5714x²
x² = 2/0.5714
x² = 3.5
x = ±√3.5 = ±1.87
So, on the graph, the coordinates of the truck at ground level are x = -1.87 and x = +1.87.
The maximum width of the load is 1.87 – (-1.87) =[tex]\boxed{\textbf{3.74 m}}[/tex]
In the diagram below, the red parabola represents the underpass, and the blue box is the truck going through it.
