Radii of stacked circles inside the graph of y = |x|^1.5

songoku
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Homework Statement
Please see below
Relevant Equations
Not sure
1684204234937.png

1684204371854.png
(a) The hint from question is to used geometrical argument. From the graph, I can see ##r_1+r_2=c_2-c_1## but I doubt it will be usefule since the limit is ##\frac{r_2}{r_1} \rightarrow 1##, not in term of ##c##.

I also tried to calculate the limit directly (not using geometrical argument at all).

$$\lim_{\frac{r_2}{r_1} \rightarrow 1} \left(\frac{r_1+r_2}{{r_2}^{1.5}-{r_1}^{1.5}}\right)$$
$$=\lim_{\frac{r_2}{r_1} \rightarrow 1} \left(\frac{1+\frac{r_2}{r_1}}{\frac{{r_2}^{1.5}}{r_1}-{r_1}^{0.5}}\right)$$

Then got stuck

I also tried rationalization:
$$\lim_{\frac{r_2}{r_1} \rightarrow 1} \left(\frac{r_1+r_2}{{r_2}^{1.5}-{r_1}^{1.5}}\right) \times \frac{{r_2}^{1.5}+{r_1}^{1.5}}{{r_2}^{1.5}+{r_1}^{1.5}}$$
$$=\lim_{\frac{r_2}{r_1} \rightarrow 1} \left(\frac{(r_1+r_2)({r_2}^{1.5}+{r_1}^{1.5})}{{r_2}^{3}-{r_1}^{3}}\right)$$
$$=\lim_{\frac{r_2}{r_1} \rightarrow 1} \left(\frac{(r_1+r_2)({r_2}^{1.5}+{r_1}^{1.5})}{(r_2-r_1)({r_2}^{2}+r_1 r_2+{r_1}^{2}})\right)$$

Then stuck again

Please give me hint, especially how to use geometrical argument.

Thanks
 
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##r_2/r_1 \rightarrow 1## is realized when ##r_1,r_2 \rightarrow +\infty## where the curve is almost vertical and the center-center distance on y axis is
r_1+r_2 \approx r_2^{1.5}-r_1^{1.5}
 
anuttarasammyak said:
##r_2/r_1 \rightarrow 1## is realized when ##r_1,r_2 \rightarrow +\infty## where the curve is almost vertical and the center-center distance on y axis is
r_1+r_2 \approx r_2^{1.5}-r_1^{1.5}
I understand.

For (b), I again tried rationalization but stuck. Do we also use geometrical argument to solve (b)?

Thanks
 
It seems to work that
r_2^{1.5}-r_1^{1.5}=\sqrt{r_2}^3-\sqrt{r_1}^3=(\sqrt{r_2}-\sqrt{r_1})(r_2+r_1+\sqrt{r_2r_1})
 
Hint:

Show that ##\displaystyle{\lim_{n \to \infty}\dfrac{r_{n+1}}{r_n}}=1## in case ##(c)## is true so that we can use the formula in ##(c)## instead.

Then show ##(c) \Longrightarrow (b) \Longrightarrow (a)## so we only have to prove ##(c)##.

Do you know any methods to prove ##(c)##?

Edit: I get ##3## as the limit in ##(a)##.
 
Last edited:
fresh_42 said:
Edit: I get ##3## as the limit in ##(a)##.
But using method in post#2, the answer is 1

fresh_42 said:
Hint:

Show that ##\displaystyle{\lim_{n \to \infty}\dfrac{r_{n+1}}{r_n}}=1## in case ##(c)## is true so that we can use the formula in ##(c)## instead.

Then show ##(c) \Longrightarrow (b) \Longrightarrow (a)## so we only have to prove ##(c)##.

Do you know any methods to prove ##(c)##?
Actually, I tried to prove (c) by using (b):

##\lim_{\frac{r_2}{r_1} \rightarrow 1}## is the same as saying ##r_2## and ##r_1## are large so for large value of ##r##, ##\sqrt{r_2}-\sqrt{r_1}\approx \frac{2}{3}##

Then I changed it into:
$$\sqrt{r_{n+1}}-\sqrt{r_n}\approx \frac{2}{3}$$
$$r_{n+1}\approx \left(\frac{2}{3}+\sqrt{r_n}\right)^2$$
$$r_{n}\approx \left(\frac{2}{3}+\sqrt{r_{n-1}}\right)^2$$

Then I don't know what to do to get (c)
 
songoku said:
But using method in post#2, the answer is 1
This isn't a method. It's a heuristic at best. I get with the use of ##(c)## and for the sake of less typing with ##x=r_n## and ##y=r_{n+1}##
\begin{align*}
\dfrac{y+x}{y\sqrt{y}-x\sqrt{x}}&=\dfrac{y+x}{(y+x)(\sqrt{y}-\sqrt{x})-x\sqrt{y}+y\sqrt{x}}\\[6pt]
&=\dfrac{1}{(\sqrt{y}-\sqrt{x})- \dfrac{x}{y+x}\sqrt{y}+\dfrac{y}{y+x}\sqrt{x}}\\[6pt]
&\stackrel{(y/x)\to 1}{\longrightarrow }\dfrac{1}{\sqrt{y}-\sqrt{x}-(1/2)\sqrt{y}+(1/2)\sqrt{x}}\\[6pt]
&\stackrel{(y/x)\to 1}{\longrightarrow }2\dfrac{1}{\sqrt{y}-\sqrt{x}}\stackrel{(y/x)\to 1}{\longrightarrow }3
\end{align*}
Not sure whether I made a mistake.

songoku said:
Actually, I tried to prove (c) by using (b):

##\lim_{\frac{r_2}{r_1} \rightarrow 1}## is the same as saying ##r_2## and ##r_1## are large so for large value of ##r##, ##\sqrt{r_2}-\sqrt{r_1}\approx \frac{2}{3}##

Then I changed it into:
$$\sqrt{r_{n+1}}-\sqrt{r_n}\approx \frac{2}{3}$$
$$r_{n+1}\approx \left(\frac{2}{3}+\sqrt{r_n}\right)^2$$
$$r_{n}\approx \left(\frac{2}{3}+\sqrt{r_{n-1}}\right)^2$$

Then I don't know what to do to get (c)
That doesn't work that way.

##(b)## is also easy with ##(c)##:
\begin{align*}
\lim_{n \to \infty}\left(\sqrt{r_{n+1}}-\sqrt{r_n}\right)&=\lim_{n \to \infty}\left(\dfrac{2}{3}(n-1)+\dfrac{2}{3}+\sqrt{r_1}-\dfrac{2}{3}(n-1)-\sqrt{r_1}\right)=\dfrac{2}{3}
\end{align*}
So it remains to show that ##\lim_{n \to \infty}\left(\sqrt{r_{n+1}}-\sqrt{r_n}\right)=\lim_{(r_2/r_1) \to 1}\left(\sqrt{r_{2}}-\sqrt{r_1}\right).## As I understand the limit on the right, they are the same. But maybe that is my mistake. What does ##r_2/r_1 \longrightarrow 1## mean? ##r_2## is a function of ##r_1## isn't it, so they are not independent.
 
fresh_42 said:
This isn't a method. It's a heuristic at best. I get with the use of (c) and for the sake of less typing with x=rn and y=rn+1
y+xyy−xx=y+x(y+x)(y−x)−xy+yx=1(y−x)−xy+xy+yy+xx⟶(y/x)→11y−x−(1/2)y+(1/2)x⟶(y/x)→121y−x⟶(y/x)→13
Not sure whether I made a mistake.
I tried it a different way referring my post #4
\frac{y+x}{y^{1.5}-x^{1.5}}=(\sqrt{y/x}-1)^{-1}\sqrt{x}^{-1}(1+\frac{\sqrt{y/x}}{1+y/x})^{-1}
In the limit RHS first coefficient goes to infinity, the second goes to zero because geometry requires x ##\rightarrow \infty## to pursuit the limit, and the third goes to 2/3.
 
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Likes songoku and fresh_42
From geometry I observe
Relation of t >0 and r, evaluating line equation and length of r,
t=-\frac{2}{9}+\sqrt{\frac{4}{81}+r^2}
Recurrence formula with t from ##c_2-c_1=r_1+r_2##
c_2-r_2=c_1+r_1thus
3t_{n+1}^{3/2}-\sqrt{13}t_{n+1}+6t_{n+1}^{1/2}=3t_{n}^{3/2}+\sqrt{13}t_{n}+6t_{n}^{1/2}
if I do not make mistakes.
1684327783625.png
 
Last edited:
  • #10
anuttarasammyak said:
I tried it a different way referring my post #4
\frac{y+x}{y^{1.5}-x^{1.5}}=(\sqrt{y/x}-1)^{-1}\sqrt{x}^{-1}(1+\frac{\sqrt{y/x}}{1+y/x})^{-1}
In the limit RHS first coefficient goes to infinity, the second goes to zero because geometry requires x ##\rightarrow \infty## to pursuit the limit, and the third goes to 2/3.
I think the problem is not in post#4 but post#2 because @fresh_42 got 3 (which is also different from question). I follow his working and it also makes sense.

fresh_42 said:
That doesn't work that way.

##(b)## is also easy with ##(c)##:
\begin{align*}
\lim_{n \to \infty}\left(\sqrt{r_{n+1}}-\sqrt{r_n}\right)&=\lim_{n \to \infty}\left(\dfrac{2}{3}(n-1)+\dfrac{2}{3}+\sqrt{r_1}-\dfrac{2}{3}(n-1)-\sqrt{r_1}\right)=\dfrac{2}{3}
\end{align*}
So it remains to show that ##\lim_{n \to \infty}\left(\sqrt{r_{n+1}}-\sqrt{r_n}\right)=\lim_{(r_2/r_1) \to 1}\left(\sqrt{r_{2}}-\sqrt{r_1}\right).## As I understand the limit on the right, they are the same. But maybe that is my mistake. What does ##r_2/r_1 \longrightarrow 1## mean? ##r_2## is a function of ##r_1## isn't it, so they are not independent.
I got (c) but not sure whether my method is valid. I used something like telescoping

$$\sqrt{r_n}-\sqrt{r_{n-1}}=\frac{2}{3}$$
$$\sqrt{r_{n-1}}-\sqrt{r_{n-2}}=\frac{2}{3}$$
$$\sqrt{r_{n-2}}-\sqrt{r_{n-3}}=\frac{2}{3}$$
$$.$$
$$\sqrt{r_2}-\sqrt{r_1}=\frac{2}{3}$$

Adding all the above:
$$\sqrt{r_n}-\sqrt{r_1}=\frac{2}{3}(n-1)$$
$$r_n=\left(\frac{2}{3}(n-1)+\sqrt{r_1} \right)^2$$

But the question states "for large r" while I also included small r so I am not sure
 
  • #11
songoku said:
I think the problem is not in post#4 but post#2 because @fresh_42 got 3 (which is also different from question). I follow his working and it also makes sense.
The fraction, thanks to post #7
1684457893933.png
=\sqrt{x}^{-1} \frac{1+a}{a\sqrt{a}-1}
=\sqrt{x}^{-1} \frac{1+a}{(\sqrt{a}-1)(a+\sqrt{a}+1)} or
=(\sqrt{y}-\sqrt{x})^{-1} \frac{1+a}{a+\sqrt{a}+1}
where
a=\frac{y}{x}
Say the fraction has limit c for a ##\rightarrow## 1
\sqrt{y}-\sqrt{x}\ \ \rightarrow \ \frac{2}{3c}
 

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