Math Quiz: Parabola Path

[Ether]

Originally Posted by Ether

Hint#2: the top and bottom boundary curves are not true quadratics.

Graph of upper and lower boundary curves

[RyanCahoon]

Can I give the answers in parametric form?

Spoiler for parametric plot equations:

x_lower(t) = t - 0.0866 t / √(1 + (0.0866 t)²)
y_lower(t) = -0.0433 t² - 1 / √(1 + (0.0866 t)²)

x_upper(t) = t + 0.0866 t / √(1 + (0.0866 t)²)
y_upper(t) = -0.0433 t² + 1 / √(1 + (0.0866 t)²)

[Ether]

Originally Posted by RyanCahoon

Can I give the answers in parametric form?

You got it

Reps to you !

[Ether]

Is it possible to represent Ryan's parametric equations in explicit form y=f(x) ?

Or even implicit f(x,y)=0 ?

[Ether]

Originally Posted by Ether

Is it possible to represent Ryan's parametric equations in explicit form y=f(x) ?

Or even implicit f(x,y)=0 ?

Does anyone have access to Mathematica?

[cgmv123]

Originally Posted by Ether

Does anyone have access to Mathematica?

http://www.wolfram.com/mathematica/trial/

[Ether]

Originally Posted by cgmv123

http://www.wolfram.com/mathematica/trial/

Thank you, I am aware of that. For a variety of reasons, I don't install or use trial software.

Are there any Mathematica gurus out there in CD land?

[maths222]

As a function of x, it will describe the upper boundary on one side and the lower on the other. It does exist, but it is very ugly. I will try to remember to post it later.

[Ether]

Originally Posted by maths222

As a function of x, it will describe the upper boundary on one side and the lower on the other.

Yes, there will be 2 separate functions - one for the upper boundary and one for the lower (just like Ryan's parametric equations).

It does exist, but it is very ugly. I will try to remember to post it later.

Excellent. Would you please post both a "typeset" version and a "cut-and-paste programming" version? (see example in attachment)

Thank you.

[maths222]

The display ones will come later.

Eq 1:

−11.547344110855*sqrt(((x^(2)+133.34115601449)/(2.*x*sqrt(x^(2)+133.34115601449)*abs(x)+x^(4)+267.68231202898*x^(2)+17779.86388728)))*sign(abs(x)+x*sqrt(x^(2)+133.34115601449))-((0.0433*(abs(x)+x*sqrt(x^(2)+133.34115601449))^(2))/(x^(2)+133.34115601449))

Eq 2:

(((11.547344110855*x*(x^(2)+133.34115601449)-11.547344110855*sqrt(x^(2)+133.34115601449)*abs(x))*sqrt(((1)/(−2.*x*sqrt(x^(2)+133.34115601449)*abs(x)+x^(4)+267.68231202898*x^(2)+17779.86388728))))/(abs(abs(x)-x*sqrt(x^(2)+133.34115601449))))-((0.0433*(abs(x)-x*sqrt(x^(2)+133.34115601449))^(2))/(x^(2)+133.34115601449))

[Ether]

Originally Posted by maths222

The display ones will come later.

Eq 1:

−11.547344110855*sqrt(((x^(2)+133.34115601449)/(2.*x*sqrt(x^(2)+133.34115601449)*abs(x)+x^(4)+267.68231202898*x^(2)+17779.86388728)))*sign(abs(x)+x*sqrt(x^(2)+133.34115601449))-((0.0433*(abs(x)+x*sqrt(x^(2)+133.34115601449))^(2))/(x^(2)+133.34115601449))

Eq 2:

(((11.547344110855*x*(x^(2)+133.34115601449)-11.547344110855*sqrt(x^(2)+133.34115601449)*abs(x))*sqrt(((1)/(−2.*x*sqrt(x^(2)+133.34115601449)*abs(x)+x^(4)+267.68231202898*x^(2)+17779.86388728))))/(abs(abs(x)-x*sqrt(x^(2)+133.34115601449))))-((0.0433*(abs(x)-x*sqrt(x^(2)+133.34115601449))^(2))/(x^(2)+133.34115601449))

Hmm.

Your equations differ from Ryan's by about 0.01 near x=5

Also, the x-intercept of your upper-boundary equation differs from Ryan's by about 0.03

Given the number of decimal places in your equations, I would have expected them to be closer.

[Ether]

Jacob,

Can you re-run Mathematica using the following parametric equations instead?

x_lower = t + 2*a*t / sqrt(1 + (2*a*t)^2);

y_lower = a*t^2 - 1 / sqrt(1 + (2*a*t)^2);


x_upper = t - 2*a*t / sqrt(1 + (2*a*t)^2);

y_upper = a*t^2 + 1 / sqrt(1 + (2*a*t)^2);

[maths222]

I used a TI-Nspire and logic to arrive at my equations, so they are probably wrong. I will take a look at them this weekend.

[Ether]

Are there any Mathematica gurus out there?

Wondering if there is a solution for the following problem:

Find explicit functions y=y_lower(x) and y=y_upper(x) for the following parametric equations:

x_lower = t + 2*a*t / sqrt(1 + (2*a*t)^2);

y_lower = a*t^2 - 1 / sqrt(1 + (2*a*t)^2);


x_upper = t - 2*a*t / sqrt(1 + (2*a*t)^2);

y_upper = a*t^2 + 1 / sqrt(1 + (2*a*t)^2);