How High Must a Parabolic Arch Be for Minimum Clearance Over a Stream?

[ahmadmz]

Homework Statement

A parabolic arch spans a stream 200 feet wide. How high
must the arch be above the stream to give a minimum
clearance of 40 feet over a channel in the center that is
120 feet wide?

Homework Equations

The Attempt at a Solution

They are asking for the k value right?
If I place the parabola starting from x=0,y=0 then the vertex is at (100,40+n).
When x=40,y=40. x=160,y=40 I want to find out what y is when x=100.

So I used the equation (x-h)^2 = a(y-k)

(x-100)^2 = a(y-(40+n)) After using x=40 ,y=40 I get n=-3600/a And now I have no idea what to do .. Am I doing this right?

 

[Dick]

I think so. Why not just put x=0 and y=0 into your equation and get another relation between 'a' and 'n'?

 

[blochwave]

I think you can just reason it out, no need for all those crazy equations and what have you

Basically you're going to have a parabola that crosses the x-axis at x=-100 and x=100

You need at least y=40 at x=60 and -60, that ultimately what it's asking, right? And it's basically of the form Y=-Ax^2+H (I just centered it at 0 for ease)

H is what we're looking for, and A is also unknown, however we know if y=40, then x=60, and if y=0, x=100

I went ahead and finished this out to find A and H and checked it and it was right, see if you can follow.

 

[ahmadmz]

Ok I used the equation y=-ax^2+h with the values x=60,y=40 and x=100,y=0.
Then solved the system and got a = -1/160 and h = 62.5 (the answer for the question).
And the equation of the parabola is y=-1/160x^2+62.5

Thanks for the help :). This is the first time I've done a question like this and I had trouble approaching it. I'm self studying these days and this shows i got a long way to go.

 

[rocomath]

Thanks for the help :). This is the first time I've done a question like this and I had trouble approaching it. I'm self studying these days and this shows i got a long way to go.

I feel your pain, I been self-studying for months now. I'm proud of you :-] Keep at it!

 

[symbolipoint]

I performed a process somewhat like what was described in #3 and #4, but obtained a somewhat different result. The standard form of a parabola was still necessary.

Do you know of any other interesting problems like this; using parabolas or other conic sections? Interesting and varied problems are often difficult to find. Also, I'm curious; was the question in post #1 from PreCalculus, or was it from Intermediate Algebra (I suspect it is from PreCalculus). Once in a while, I restudy College Algebra or Intermediate Algebra, and the applied situation exercises are often interesting but I just do not find enough of them.

 

[ahmadmz]

What result did you get symbolipoint? The book says the answer is 62.5 which i got.
I am studying conics right now and I'll let you know if I find any more similar problems. This is from a intermediate algebra book. You are correct that most books do not have many problems like these. They are indeed interesting :)

 

[symbolipoint]

What result did you get symbolipoint? The book says the answer is 62.5 which i got.
I am studying conics right now and I'll let you know if I find any more similar problems. This is from a intermediate algebra book. You are correct that most books do not have many problems like these. They are indeed interesting :)

I just wrote a lengthy response and the forum cut me off.

In short, \[<br /> f(x) = - \frac{1}{{90}}x^2 + 40 + 71{\textstyle{1 \over 9}}<br /> \]<br />

I used an "untranslated" parabola, and then a "translated" parabola. Too difficult to rewrite all the details NOW. One used x=60, the other relied on x=100.

 

[ahmadmz]

Hmm I think you made a mistake somewhere. Can someone confirm the answer please? What did you get blochwave?

 

[symbolipoint]

I think you can just reason it out, no need for all those crazy equations and what have you

Basically you're going to have a parabola that crosses the x-axis at x=-100 and x=100

You need at least y=40 at x=60 and -60, that ultimately what it's asking, right? And it's basically of the form Y=-Ax^2+H (I just centered it at 0 for ease)

H is what we're looking for, and A is also unknown, however we know if y=40, then x=60, and if y=0, x=100

I went ahead and finished this out to find A and H and checked it and it was right, see if you can follow.

Blockwave understood the problem description. I may have obtained the "wrong" answer because I did not fully understand the problem description. Yet, he seems to have taken most of the approach that I took. Are we both misunderstanding something?