SOLUTION: the parabolic opening of a tunnel is 32m wife measured from side to side along the ground. at the points that are 4m from each side the tunnel entrance is 6 m high determine

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Question 1100625: the parabolic opening of a tunnel is 32m wife measured from side to side along the ground. at the points that are 4m from each side the tunnel entrance is 6 m high

determine a equation of the function that models the opening of the tunnel
Found 2 solutions by Fombitz, ikleyn:
Answer by Fombitz(32388) About Me  (Show Source):
You can put this solution on YOUR website!
So set a coordinate system in the middle of the tunnel at (0,0).
The top of the tunnel would be (0,6), the vertex of the parabola.
y=a%28x-0%29%5E2%2B6
y=ax%5E2%2B6
The two other points are the width of the tunnel at (-16,0) and (16,0).
Plugging that into the equation,
0=a%2816%29%5E2%2B6
256a%2B6=0
256a=-6
a=-6%2F256
a=-3%2F128
highlight%28y=-%283%2F128%29x%5E2%2B6%29
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Answer by ikleyn(52852) About Me  (Show Source):
You can put this solution on YOUR website!
.
the parabolic opening of a tunnel is 32 m wide measured from side to side along the ground.
at the points that are 4 m from each side the tunnel entrance is 6 m high

determine a equation of the function that models the opening of the tunnel
~~~~~~~~~~~~~~~~~~ 

Put the origin of the coordinate system at the highest point of the arch, the axis "y" directed vertically up.

so the the origin of the coordinate system will be THE VERTEX of the parabola.


You have y = ax^2 as the equation of the parabola with the negative unknown coefficient "a".


The equation for "a" is

y(12) - y(16) = a%2A12%5E2 - a%2A16%5E2 = 6,     (1)

saying that  at the distance of 12 = 16-4 meters from the side the height of the tunnel is 6 meters.


Then  a = 6%2F%2812%5E2-16%5E2%29 = 6%2F%28144-256%29 = -6%2F112 = -3%2F56.


Hence and finally, the equation of the parabola is

y = %28-3%2F56%29%2Ax%5E2 

in this coordinate system.   // And, although the problem does not ask about it, the height of the tunnel is  %283%2F56%29%2A16%5E2 = 13.71 meters.

The tutor @Fombitz incorrectly interpreted the condition.