I'm stumped..

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December 8th 2009, 09:21 PM
NKS

I'm stumped..

A bridge is built in the shape of a parabolic arch. The bridge arch has a span of 196 feet and a max height of 35 feet. Find the height of the arch at 25 feet from its center.

I've done this problem now about 5 or 6 times.. and the answer I get is different almost every time. I wish I could draw it to show what my thoughts on the set up are. basically I did y=ax^2 first but then thought I needed to find P. so that would be y=1/4p(x^2). somewhere along the lines I'm almost positive I just confused myself and did something wrong. I ended up with about 25.8. The other problems in this same vein I have been able to understand and figure out because of help from here, but this one has me stumped.. maybe its the wording or something. maybe its just late. help would be appreciated.
December 8th 2009, 11:26 PM
Prove It

Quote:

Originally Posted by NKS

A bridge is built in the shape of a parabolic arch. The bridge arch has a span of 196 feet and a max height of 35 feet. Find the height of the arch at 25 feet from its center.

I've done this problem now about 5 or 6 times.. and the answer I get is different almost every time. I wish I could draw it to show what my thoughts on the set up are. basically I did y=ax^2 first but then thought I needed to find P. so that would be y=1/4p(x^2). somewhere along the lines I'm almost positive I just confused myself and did something wrong. I ended up with about 25.8. The other problems in this same vein I have been able to understand and figure out because of help from here, but this one has me stumped.. maybe its the wording or something. maybe its just late. help would be appreciated.

The horizontal distance is 196 feet.

So if you assume it starts at the origin, then two points you have are

$(0, 0)$ and $(196, 0)$ .

You also have the maximum height of 35 feet. This occurs right in the centre of your x-intercepts.

So the turning point is $(98, 35)$ .

If you use the turning point form for the equation of the parabola, it will be of the form

$y = a(x - h)^2 + k$ .

Substituting in the turning point gives

$y = a(x - 98)^2 + 35$

And substituting in another point, in this case $(196, 0)$ gives you the a value.

$0 = a(196 - 98)^2 + 35$

$0 = a(98)^2 + 35$

$a = -\frac{35}{98^2}$ .

Therefore the equation of the parabola is

$y = -\frac{35}{98^2}(x - 98)^2 + 35$ .

Now use it to find the height when $x = 25$ .
December 9th 2009, 09:55 AM
NKS

I think, in the final form of the equation that you found, the 98 should be a 0. because (h,k) is the vertex, no? and the vertex, if are drawing this out to resemble a bridge, would be at (0,35). I also change the h to 0 in the beginning and followed your steps and came to the same answer as I did after I had just changed it in the final equation. so the answer should be 32.7 I believe.

maybe someone could check to see if I am right.
December 9th 2009, 12:45 PM
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Essentially it doesn't matter where you put the origin, as it will just end up creating a translation of the graph.

I always put the origin as the first known point - in this case, where the bridge starts...