A tunnel with a parabolic arch is 12m wide. DID I SOLVE THIS CORRECTLY? PLEASE, TELL/HELP ME!

Question

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

My answer

y = a (x – r) (x – s)
y = a (x - 12) (x – 0)
y = a (4 – 12) (4 – 0)
6 = a (-8) (4)
6 = a ( - 32)
6 / - 32 = a
3 / - 16 = a

Therefore, the equation is: y = 3 / - 16 x(x + 12)

Sub y = 5

5 = 3 / - 16 x(x + 12)
5 = 3 / - 16 (x ^2+ 12x)
5 = 3 / - 16 x ^2 – 36 / 16x
5 = 3 / - 16 x ^2 – 9 / 4x
y= 3 / - 16 x ^2 – 9 / 4x – 5 = 0

Use quadratic formula:

-b + - √ b^2 – 4ac / 2a

Plug in numbers from equation (to lazy to type it all out)

(+) = - 1.89
( - ) = 13.89

My teacher told me that with this answer, the positive one –since negative can’t work—that it would fit underneath the bridge. My question is why, though. I don’t understand that one part. Isn't the bridge only 3.5 m wide underneath? And P.S, did I solve the question correctly?

Hello there.
Your method of solving the first part of the problem is very slick! Also your methodology for the rest of the problem is spot on.
However, I did notice the solutions to the quadratic equation:

(3/16)x^2 -(9/4)x - 5 = 0

You gave them as -1.89 and +13.89.
But your origin for the quadratic function is at the bottom left hand corner of the tunnel and your solutions to the above equation are for when this tunnel intersects the line y = 5, surely, I reasoned, both answers should be positive. I then checked back and noticed you gave the quadratic function as:

y = 3 / - 16 x(x + 12)

when it should have been (from your earlier work):

y = - (3 / 16) x(x - 12)

Using this results and the quadratic equation we get the x values(the working out of which I am also too lazy to write down!) for where y=5 intersects the parabola as:

x = 2.94 and 9.06