Re: Parabola word problem

Hey bearjr32.

You need to figure out the equation of the parabola which has the form f(x) ax^2 + bx + c = 0.

Hint: You have f(-200) = f(200) = 0 if x = 0 is the middle of the bridge and you also have f(0) = 16.

Can you use this to solve for a, b, and c and get the height at those arbitrary end-points?

Re: Parabola word problem

Chiro

To be honest I am 30 and haven't done anything but very basic algebra and trig since I graduated. I'm struggling even getting that far.. Really I am struggling with this whole parabola concept. I get the basic formulas and the focus and all that but after that I am lost. That's what I get for delaying my college education!

Re: Parabola word problem

No worries, I'll explain it a bit further.

A parabola has the form of y = f(x) = ax^2 + bx + c. It has a minimum or a maximum (depending on if it is concave up or down) and it is symmetric about this value (for symmetry think about that if you drew a line parallel to the y axis at the maximum or minimum, the graph would be a reflection around that line).

From the information you have given there is a maximum at 16m at the center of the bridge.

What I have done is chosen for the middle of the bridge to be at the y-axis (or at x = 0). Since the height is 16m higher than the end points we set f(0) = 16.

We also know that the span is 400m and because of the symmetry, we know that f(-x) = f(x) and since the span is 200m, we know that f(-200) = f(200) = 0.

If you graphed it, it would look like a half football with the highest point at x = 0 and the end points at x = -200 and x = 200.

Since the function of a parabola is always f(x) = ax^2 + bx + c and since we have three pieces of independent information for f(x), we can solve for the parameters a, b, and c which we have to work out.

So lets look at the pieces of information available:

f(0) = 16 means a*(0)^2 + b*(0) + c = 16 which means c = 16 (Substituting x = 0 into f(x)))

f(-200) = 0 means a*(-200)^2 + b*(-200) + c = 40000a - 200b + 16 = 0

f(+200) = 0 means a*(200)^2 + b*(200) + c = 40000a + 200b + 16 = 0

So we have c but we need to figure out a. Remember that we have two equations:

40000a - 200b + 16 = 0 (Equation 1)

40000a + 200b + 16 = 0 (Equation 2)

Adding both equations we get Equation 3:

80000a + 32 = 0 which means

a = -32/80000.

To get b we use Equation 4 = Equation 2 - Equation 1 which leaves us with:

400b = 0 which means b = 0

You could have substituted a into the equation to get b as well.

So now we have the equation for our bridge that goes from x = -200 to x = 200 with the center at x = 0 which is

f(x) = -32/80000*x^2 + 16

Now if you want the height of the bridge relative to that of the end-points, you simply substitute in a value of x (remembering that the start of the bridge starts at x = -200) and get f(x) which is the height.

Re: Parabola word problem

It's amazing when someone shows it to me how simple it seems. Do you mind checking me on this next one? The shape of a wire hanging between two poles closely approximates a parabola. Find the equation of a wire that is suspended between two poles 40m apart and whose lowest point is 10m below the level of the insulator. I came up with the formula

F(x)=1/40x^2

Re: Parabola word problem

Show us your working if you don't mind.

Re: Parabola word problem

I did the three equations again

F(x)=ax^2+bx+c

using the lowest point as origin it gives the first coordinate (0,0) then I know that (-20,10) and (20,10) are the endpoints

F(0)=a(0)^2+b(0)+c=0 C=0

F(-20)=400a-40b+0=10

F(20)=400a+40b+0=10

adding the bottom two together gives me 800a=20 or a=1/40

substituting this into an equation gives me 10+40b=10 or b=0

knowing b and c are equal to zero I am left with F(x)=1/40(x^2)

Re: Parabola word problem