SOLUTION: Okay, so I am stuck with Paul's truck company which hauls oversized loads. The trucks travel through a tunnel with an opening in the shape of a parabola that is 12 meters wide and

Algebra ->  Rectangles -> SOLUTION: Okay, so I am stuck with Paul's truck company which hauls oversized loads. The trucks travel through a tunnel with an opening in the shape of a parabola that is 12 meters wide and       Log On


   

Question 826459: Okay, so I am stuck with Paul's truck company which hauls oversized loads. The trucks travel through a tunnel with an opening in the shape of a parabola that is 12 meters wide and 6 meters high.
I am told to find a function, T(x), for the shape of the tunnel...I am also supposed to graph it but I don't think that would be much of a trouble if I get a walk-in thus far.
I appreciate your help in advance, and I will be looking forward to your reply soon. Thanks a bunch! :)
Answer by josmiceli(19441) About Me  (Show Source):
You can put this solution on YOUR website!
I can say that this is a parabola that looks like a bowl
turned upside down. To simplify, say it is symmetric
about the origin. That means it intersects the x-axis
at ( 6,0 ) and ( -6,0 ) The T-intercept is at ( 0,6 ).
-------------------------------------------
The general form of the parabola is:
+T%28x%29+=+-a%2Ax%5E2+%2B+b%2Ax+%2B+c+
The formula for the vertex is:
+x%5Bv%5D+=+-b%2F%282a%29+
Since the vertex is on the y-axis,
+x%5Bv%5D+=+0+
I can say:
+T%28x%29+=+-a%2A0%5E2+%2B+b%2A0+%2B+c+
+y%5Bv%5D+=+6+
+6+=+c+
-----------
So far, I have:
+T%28x%29+=+-a%2Ax%5E2+%2B+b%2Ax+%2B+6+
Plug in ( 6,0 )
+0+=+-a%2A6%5E2+%2B+b%2A6+%2B+6+
+-36a+%2B+6b+%2B+6+=+0+
+-6a+%2B+b+%2B+1+=+0+
(1) +-6a+%2B+b+=+-1+
also
Plug in (-6,0 )
+T%28x%29+=+-a%2A%28-6%29%5E2+%2B+b%2A%28-6%29+%2B+6+
+0+=+-36a+-+6b+%2B+6+
+-36a+-+6b+%2B+6+=+0+
+-6a+-+b+%2B+1+=+0+
(2) +-6a+-+b+=+-1+
Add (1) and (2)
(1) +-6a+%2B+b+=+-1+
(2) +-6a+-+b+=+-1+
+-12a+=+-2+
+a+=+1%2F6+
-------------
Plug this back in:
(1) +-6%2A%281%2F6%29+%2B+b+=+-1+
(1) +-1+%2B+b+=+-1+
(1) +b+=+0+
--------------
So, the equation is:
+T%28x%29+=+-%281%2F6%29%2Ax%5E2+%2B+6+=+0+
Here's the plot:
+graph%28+400%2C+400%2C+-8%2C+8%2C+-2%2C+8%2C+-%281%2F6%29%2Ax%5E2+%2B+6+%29+