Transformation of a equation: L = pi((D+d)/2) + S(sqrt[S^2+(D-d)/2)])

[Deazfish]

Hi,

I am working on a textbook-problem which I can't solve, and was hoping you could help me.
The problem is to show that the equation:

(1) L = π((D+d)/2)+2(sqrt(S^2+((D-d)/2))^2)

Is the same as:

(2) L = π((D+d)/2) + 2S + (1/S)((D-d)/2))

The task has 2 hits:
(i) Sqrt(a^2+x^2) = a*sqrt(1+(x/a)^2)
(ii) Sqrt(1+t) = 1+ t/2 if t is small enough

I keep getting the result:
L = π((D-d)/2) + 2S + (1/S)((D+d)/2))^2

Could someone point out my mistake?

Thank you in advance.

 

[stapel]

The problem is to show that the equation:

(1) L = π((D+d)/2)+S(sqrt(S^2+((D-d)/2)))

Is the same as:

(2) L = π((D+d)/2) + 2S + (1/S)((D-d)/2))

To be clear, do you mean the following?

. . . . .\(\displaystyle \mbox{(1) }\, L\, =\, \pi \left(\dfrac{D\, +\, d}{2}\right)\, +\, S \left(\sqrt{\strut S^2\, +\, \dfrac{D\, -\, d}{2}\,}\right)\)

. . . . .\(\displaystyle \mbox{(2) }\, L\, =\, \pi \left(\dfrac{D\, +\, d}{2}\right)\, +\, 2S\, +\, \left(\dfrac{1}{S}\right) \left(\dfrac{D\, -\, d}{2}\right)\)

The task has 2 hits:

Do you perhaps mean "hints"? If not, what is meant by "hits"?

(i) Sqrt(a^2+x^2) = a*sqrt(1+(x/a)^2)
(ii) Sqrt(1+t) = 1+ t/2 if t is small enough

What does this mean when it says "if t is small enough"? I mean, if you were taking a limit or something, then maybe the size of a variable might matter, but not in general.

I keep getting the result:
L = π((D-d)/2) + 2S + (1/S)((D+d)/2))^2

Could someone point out my mistake?

After you show your work, so we know how you arrived at this result, I'm sure somebody will be glad to try. Thank you!

 

[Deazfish]

Sorry, couple of errors in the first post.

I am supposed to show that the equation:
[FONT=MathJax_Main](1) L = π((D+d)/2)+2(sqrt(S^2+((D-d)/2))^2) (the S before sqrt is supposed to be a 2)[/FONT]

Is the same as:

(2) L = π((D+d)/2) + 2S + (1/S)((D-d)/2)) (your equation is right)

With the hints that:
(i) Sqrt(a^2+x^2) = a*sqrt(1+(x/a)^2)
(ii) It can be shown that if x is small enough, then: Sqrt(1+t) = 1+ t/2

I attach a photo of the original problem (sorry, but it is in Norwegian, but you should be able to get an idea of the equations). I am new to the forum, so I don't know how to use the forumlas, so I have attached "my solution by hand" as a photo to show my work.
I keep getting the result:

(2) L = π((D+d)/2) + 2S + (1/S)((D-d)/2))^2

I think the textbook is wrong, because the (1/s)((D-d)/2) is unitless and can not be added to 2S or Pi((D+d)/2). I also plotted the problem with values for D (0,12 cm) d (0,6 cm), and S (75cm) and my equation is closer (but not accurate, because of the approximation in hint (ii)) than the equation the textbook provides.

The pictures came on their side

 

[mmm4444bot]

Let k = (D - d)/2

Before: 2SQRT(S^2 + k)

I think you meant

Before: 2SQRT(S^2 + k^2)

So you want to show that before = after, right?

Hint (ii) implies that before ≈ after. :cool:

 

[mmm4444bot]

The problem is to show that the equation:

(1) L = π((D+d)/2)+2(sqrt(S^2+((D-d)/2)))

is the same as:

(2) L = π((D+d)/2) + 2S + (1/S)((D-d)/2))

They're not asking you to show that equation (1) is the same as equation (2).


They're asking you to show how equation (1) can be approximated by equation (2), if [(D-d)/(2S)]^2 is sufficiently small.


The task has 2 [hints]:

(i) Sqrt(a^2+x^2) = a*sqrt(1+(x/a)^2)

(ii) Sqrt(1+t) = 1+ t/2 if t is small enough (ii) has no = sign

The symbol ≈ is read as "approximately equal to". :cool:

I keep getting the result:

L = π((D-d)/2) + 2S + (1/S)((D+d)/2))^2

Could someone point out my mistake?

I think your result is correct.

I think the book contains a typographical error. The exponent 2 (highlighted in blue above) is missing.

PS: In the future, please rotate and crop your images before posting. You can see how your post will appear (for proofreading), by using the Preview Post button before submitting. ~ Cheers

 

[mmm4444bot]

No, I didn't:
the 1st post did not originally contain the ^2;
I see it was edited after Stapel's and my post.

Maybe you didn't see the post right before yours.

 

[mmm4444bot]

Any reason for adding the ^2?

See post #3. There's an image of the book.

I think (D-d)/2 ought to be squared in equation (2), also. I think it's a typo.