Probability of a triangle

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June 23rd 2006, 09:30 PM
malaygoel

Quote:

Originally Posted by JakeD

$\begin{aligned} x + y &\ge 1 \\ x^2 + y^2 &\le 1 \\ 0 \le x &\le 1 \\ 0 \le y &\le 1. \\ \end{aligned}$

I tried this but could not get much far.
$x=cosA (0<A<\frac{\pi}{2})$
$y=cosB (0<B<\frac{\pi}{2})$
$cosA + cosB \leq 1 \leq cos^2A + cos^2B$

KeepSmiling
Malay
June 24th 2006, 02:05 AM
JakeD

Quote:

Originally Posted by JakeD

Use integral calculus to find the area of the region

$\begin{aligned} x + y &\ge 1 \\ x^2 + y^2 &\le 1 \\ 0 \le x &\le 1 \\ 0 \le y &\le 1. \\ \end{aligned}$

Call this region $D$ and rewrite it as

$\begin{aligned} 0 \le x &\le 1 \\ 1-x \le y &\le \sqrt{1-x^2}. \\ \end{aligned}$

The probability, which is a double integral, can then be evaluated as an iterated integral

$\begin{aligned}P &= \underset{D\ \ }{\iint} 1\ dxdy \\ &= \int_0^1 dx\ \int_{1-x}^{\sqrt{1-x^2}} 1\ dy \\ &= \int_0^1 \sqrt{1-x^2} - (1-x)\ dx \\ &= \sin^{-1}(1)/2 - (1 - 1/2) \\ &= \frac{\pi -2}{4}. $

I used The Integrator to find the antiderivative of $\sqrt{1-x^2} .$
June 24th 2006, 09:52 PM
malaygoel

Quote:

Originally Posted by JakeD

Call this region $D$ and rewrite it as

$\begin{aligned} 0 \le x &\le 1 \\ 1-x \le y &\le \sqrt{1-x^2}. \\ \end{aligned}$

The probability, which is a double integral, can then be evaluated as an iterated integral

$\begin{aligned}P &= \underset{D\ \ }{\iint} 1\ dxdy \\ &= \int_0^1 dx\ \int_{1-x}^{\sqrt{1-x^2}} 1\ dy \\ &= \int_0^1 \sqrt{1-x^2} - (1-x)\ dx \\ &= \sin^{-1}(1)/2 - (1 - 1/2) \\ &= \frac{\pi -2}{4}. $

I used The Integrator to find the antiderivative of $\sqrt{1-x^2} .$

Great Work JakeD!

I want to tell you that this problem is assigned to a student with the instructions to avoid calculus and coordinate geometry for this question(I am sorry, I should have told you this before).
You can use trigonometry and combanotrics,algebra and number theory.

Keep Smiling
Malay
June 25th 2006, 02:39 AM
JakeD

Quote:

Originally Posted by malaygoel

Great Work JakeD!

I want to tell you that this problem is assigned to a student with the instructions to avoid calculus and coordinate geometry for this question(I am sorry, I should have told you this before).
You can use trigonometry and combanotrics,algebra and number theory.

Keep Smiling
Malay

What is the point of this problem? To take something very simple and make it complicated with combinatorics and number theory? Or is it to keep it simple? I think it is the latter. So I'd just go with the picture as your rough sketch and be done with it.
June 25th 2006, 09:13 PM
malaygoel

Quote:

Originally Posted by JakeD

What is the point of this problem? To take something very simple and make it complicated with combinatorics and number theory? Or is it to keep it simple? I think it is the latter. So I'd just go with the picture as your rough sketch and be done with it.

Ok
Nice
Thanks for your support

Keep smiling
Malay

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