A tunnel with a parabolic arc is 12 m wide. If the height of the arc 4m from the left edg is 6m,can a truck that is 5m.....tall and 3.5m wide pass through the tunnel? Justify your answer
You should use the following factored form of quadratic equation such that:
`y = a(x - x_1)(x - x_2)`
`x_1` and `x_2` are the solutions to quadratic equation
Since the problem provides the wide of tunnel of `12 m` , hence, you may consider the endpoints of tunnel as `x_1 = 0` and `x_2 = 12 ` such that:
`y = a(x - 0)(x - 12)`
Since the problem provides the height of 6 m, if the tunnel is 4m wide, you may substitute 6 for y and 4 for x in equation such that:
`6 = a(4 - 0)(4 - 12) => 6 = -32a => a = -6/32 => a = -3/16`
Hence, evaluating the quadratic equation that models the tunnel yields `y = (-3/16)x(x - 12).`
Since you need to check if a truck whose dimensions are 5m tall and 3.5 m wide can pass through the tunnel, you need to substitute 5 for y in the given equation such that:
`5 = (-3/16)x(x - 12) => 16*5 = -3x(x - 12)`
Opening the brackets yields:
`80 = -3x^2 + 36x`
You need to move all terms to one side, such that:
`3x^2 - 36x + 80 = 0`
You need to use quadratic formula such that:
`x_(1,2) = (-(-36)+-sqrt((-36)^2 - 4*3*80))/(2*3)`
`x_(1,2) = (36+-sqrt(1296 - 960))/6 => x_(1,2) = (36+-4sqrt21)/6`
`x_(1,2) = (18+-2sqrt21)/3 => x_1 = 9.05 ; x_2 = 2.945`
You need to check what is the distance between the points `x_1` and `x_2` such that:
`x_1 - x_2 = 9.05-2.945 = 6.11 m` (Notice that the right distance between two points `x_1` and `x_2` is of `6.11 m` , hence, this distance is the allowed distance)
Notice that if the truck is placed at the left point `x_2 = 2.945m` , its tall is larger than the tunnel tall at that distance, since the problem provides the information that the tunnel is 6 m tall at the wide of 4 m.
If the truck is centered, between the points `x_1` and`x_2` , then its width 0f 3.5m does not fit in the allowed width of `6.11/2 = 3.05m` .
Hence, evaluating if a truck of 5 m tall and 3.5 m wide can pass through the tunnel yields that is not possible.
The tunnel's parabolic arch goes through (0,0), (4,6) and (9,6). Using these points, the equation of the parabolic curve is y = -1/6x² + 13/6x.
If the truck was exactly centered going through the arch, it would go from the 4.75 meter mark to the 8.25 meter mark of the parabola. If we plug x = 4.75 or 8.25 into the parabolic curve, the height at both these positions is 6.53125 meters. This is easily enough height for a 5 m tall truck to go through.
The textbook looked to see where the parabolic curve was 5 meters tall. If 5 = -1/6x² + 13.6x, then -1/6x² + 13/6x - 5 = 0.
Multiplying this by -6 gives:
x² - 13x + 30 = 0
(x - 10)(x - 3) = 0
x = 10 and x = 3
The arch is 5 meters tall when you are 3 meters from either edge of the parabolic arch at 3 and 10 meters. That means a truck 5 meters tall can go through the arch if it is less than 7 meters wide and is driving in the center of the arch.
(I think the textbook's 6.11 m wide is wrong.)
Either way the truck can go through the arch.
i hope that helps..!! (: