How to choose initial guesses for Secant method (lateral surface area of a cone)

[PA3040D]

Dear all, this equation doesn't include initial guesses, so it cannot be solved. Could you please advise on how to incorporate initial guesses? Are there any suggested values we could propose. Can I use any arbitrary two guesses .

 

[blamocur]

Which initial guesses?
In the equation for the surface you are given values for two out of three variables -- can you figure out the value of the third variable?

 

[PA3040D]

Hi
Thanks for the response

I am asking r1 and r0 values to substitute the Secant equation

 

[Dr.Peterson]

Hi
Thanks for the response

I am asking r1 and r0 values to substitute the Secant equation

View attachment 38000

First, what are you using for f?

I'd probably first try some small integer values, looking for a place where f(x) changes sign, and use one value on each side of that change. That's where the method is most likely to converge nicely (not that I'm an expert on the method). Since the method is not guaranteed to converge even under those conditions, you could have to try another pair if it didn't work well.

Or you could use a grapher to locate places to start, which is the tech version of the same idea.

 

[blamocur]

I am confused: are your posts #1 and #3 related in some way?

 

[PA3040D]

I am confused: are your posts #1 and #3 related in some way?

Yes.....The equation says find the radius of the cone using Secant Method

 

[blamocur]

Yes.....The equation says find the radius of the cone using Secant Method

Ooops, my bad: did not read the last paragraph carefully. My apologies.

 

[mario99]

Choose any two points. The closer the points to the root, the less iterations you will need.

Start with [imath]x_0 = 0[/imath] and [imath]x_1 = 1[/imath].

 

[PA3040D]

Choose any two points. The closer the points to the root, the less iterations you will need.

Start with [imath]x_0 = 0[/imath] and [imath]x_1 = 1[/imath].

Can we use like this ?
I am confused which value is used to converge to zero
Please advise

 

[Dr.Peterson]

Can we use like this ?
I am confused which value is used to converge to zero
Please advise

View attachment 38002

Please answer my question:

First, what are you using for f?

In the secant method as I understand it, you are looking for a zero (or "root") of the function f, right? That is not true of the function you are evaluating here.

What you have found is that the solution (the zero of the function f you should be using) is between r = 3 and 4. So it will be a good idea to use those as your initial guesses.

 

[mario99]

I would listen to what Dr.Peterson said, and choose [imath]r_0 = 3[/imath] and [imath]r_1 = 4[/imath].

You have:

[imath]\displaystyle f(r) = \pi r \sqrt{r^2 + 10^2} - 100[/imath]

[imath]r_2 = r_1 - \frac{f(r_1)(r_1 - r_0)}{f(r_1)-f(r_0)}[/imath]

Just make sure this is the secant approximation formula.

 

[PA3040D]

Please answer my question:

In the secant method as I understand it, you are looking for a zero (or "root") of the function f, right? That is not true of the function you are evaluating here.

What you have found is that the solution (the zero of the function f you should be using) is between r = 3 and 4. So it will be a good idea to use those as your initial guesses.

Yes Sir I just understood Thank you so much

 

[PA3040D]

Hoped this is the answer for the question please check if you have free time

 

[blamocur]

You can plug in the computed value of [imath]r[/imath] and see if you get the correct [imath]S(r)[/imath]. Or you can solve for [imath]r[/imath] explicitly:
[math]S = \pi r \sqrt{r^2 + h^2} \rightarrow S^2 = \pi^2 r^2(r^2+h^2)[/math]For [imath]u=r^2[/imath] : [math]S^2 = \pi^2 u(u+h^2) = \pi^2 u^2 + \pi^2h^2 u[/math][math]u =\frac{-\pi^2h^2 +\sqrt{\pi^4h^4 + 4\pi^2S^2}}{2\pi^2}\approx 9.27235305286417[/math][math]r = \sqrt{u} \approx 3.045053866988919[/math]

 

[Dr.Peterson]

Hoped this is the answer for the question please check if you have free time
View attachment 38003

The work and answer look correct, though the notation could be improved. (Have you been taught to use [imath]fr_1[/imath] rather than [imath]f(r_1)[/imath], for example? And why do you write "[imath]fr_2[/imath]" where I expect "[imath]r_2[/imath]?)

I would have started by saying something like this:

Let [imath]f(r)=S(r)-100=\pi r\sqrt{r^2+10^2}-100[/imath].​

Let [imath]r_0=3[/imath] and [imath]r_1=4[/imath].​

Then [imath]f(r_0)=f(3)=\pi 3\sqrt{3^2+10^2}-100=-1.60243[/imath] and​

[imath]f(r_1)=f(4)=\pi 4\sqrt{4^2+10^2}-100=35.34395[/imath].​

Therefore, [imath]r_2=\dots[/imath]​

...​

​

 

[PA3040D]

The work and answer look correct, though the notation could be improved. (Have you been taught to use [imath]fr_1[/imath] rather than [imath]f(r_1)[/imath], for example? And why do you write "[imath]fr_2[/imath]" where I expect "[imath]r_2[/imath]?)

I would have started by saying something like this:

Let [imath]f(r)=S(r)-100=\pi r\sqrt{r^2+10^2}-100[/imath].​

Let [imath]r_0=3[/imath] and [imath]r_1=4[/imath].​

Then [imath]f(r_0)=f(3)=\pi 3\sqrt{3^2+10^2}-100=-1.60243[/imath] and​

[imath]f(r_1)=f(4)=\pi 4\sqrt{4^2+10^2}-100=35.34395[/imath].​

Therefore, [imath]r_2=\dots[/imath]​

...​

​

Grate thanks .....I must improve

 

[PA3040D]

You can plug in the computed value of [imath]r[/imath] and see if you get the correct [imath]S(r)[/imath]. Or you can solve for [imath]r[/imath] explicitly:
[math]S = \pi r \sqrt{r^2 + h^2} \rightarrow S^2 = \pi^2 r^2(r^2+h^2)[/math]For [imath]u=r^2[/imath] : [math]S^2 = \pi^2 u(u+h^2) = \pi^2 u^2 + \pi^2h^2 u[/math][math]u =\frac{-\pi^2h^2 +\sqrt{\pi^4h^4 + 4\pi^2S^2}}{2\pi^2}\approx 9.27235305286417[/math][math]r = \sqrt{u} \approx 3.045053866988919[/math]

Yes of course good idea

Expected value is 100

Funny thing is..... I thought up to now the Secant method is used for finding Root of the function

 

[blamocur]

Funny thing is..... I thought up to now the Secant method is used for finding Root of the function

It is. In your case the function is [imath]f(r) = \pi r\sqrt{h^2+r^2} - S[/imath]. Also, in your case the root can be expressed explicitly, but this is not always the case.

 

[Steven G]

Dear all, this equation doesn't include initial guesses, so it cannot be solved. Could you please advise on how to incorporate initial guesses? Are there any suggested values we could propose. Can I use any arbitrary two guesses .

View attachment 37998

You need to think a bit more clearly. You say ...so it can't be solved. Then you ask how to find initial guesses so that you can find the solution.