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

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sciencesolve | Teacher | (Level 3) Educator Emeritus

Posted on

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.

Top Answer

akshaygoel96's profile pic

akshaygoel96 | Student, Grade 11 | (Level 1) Honors

Posted on

Hi,

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..!! (:

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