Question

A tunnel with a parabolic arch is 12 m wide. If the height of the arcccch 4 m from the left edge is 6 m,can a truck that is 5 m tall and 3.5 m wide pass through the tunnel.

Answer 1

Even without figuring the equation of the parabola, this can be answered.

If the top of the arch is at point (0,k), then at x = 2 or -2 the height is 6.

A 3.5m truck driven down the center of the arch extends from x = -1.75 to 1.75

Thus, it is inside the area where we know the arch is at least 6m high. No problem.

_______________

However, if the truck is required to stay on one side of the center line, we need to know how high the arch is when x = 3.5

so, let

y = ax^2 + c

6 = 4a + c (height is 6 when x = 2)
0 = 36a + c (height is zero when x=6)

a = -3/16
c = 27/4

16y = -3x^2 + 108

If the truck is on the center line, its other side is at x=3.5

16y = -3(12.25) + 108
16y = 71.25
y = 4.45m

Truck will hit it.

For extra credit: how wide can a 5m high truck be, if it must pass under the arch
(a) in the center
(b) in its lane
?