A highway tunnel has a shape that can be modelled by the equation of a parabola. Determine the following...

Question

Question #1:

A highway tunnel has a shape that can be modelled by the equation of a parabola. The tunnel is 18 m wide and the height of the tunnel 16 m from the edge is 5 m.

a) Determine the equation of the parabola.

b) Sketch a graph of your parabola. Capture your work and insert the image or link here. :)

c) Can a truck that is 8 m tall and 4 m wide pass through the tunnel? Justify your decision.

Answer 1

tunnel is 18 meters wide

graph the parabola with axis of symmetry x =0, x intercepts x=9 and x=-9

y intercept = 5sqr7

the height of the tunnel is the y coordinate of the vertex = y intercept

h= sqr(16^2- 9^2) = sqr(256-81) = sqr175 = 5sqr7 = about 13.28 meters

vertex = (0, 5sqr7)

the equation of the parabola is y = f(x) = -x^2 + 5sqr7

a truck 8 m tall and 4m wide

will have x coordinates of -2 and 2, with height allowed of correspnding y coordinate (-2)^2 +5sqr7 = 13.28-4= 9.28.

8m is less than 9.28 meters, so the truck can fit in the tunnel if it's driving in the center of the tunnel's road

with no oncoming traffic.

it doesn't fit to the right of the roadway's center as 8=-x^2 + 13.28

x^2 =13.28-8= 5.28

x = sqr5.28 = 2.3 meters< 4 meters, leaving no leeway

the truck's x coordinates would be 0 and 4

y = -4^2 + 13.28 = -16+13.28 = 2.72 < 8